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THE THIRTEEN BOOKS OF EUCLID'S ELEMENTS * : "?^iV?-*' v ivv', J. ■* V L < -*V ^VJ * : *, JT <v Aj^ a^.* uitf Tj N* 1 1 Unfnu-i^(."fJ»*&4*rrouIifrjrrp«v>o.UO!?-t' ytX/T*-«^t «wrt>ft«o<ni» " * * * ' * ■ • • »* m ■«• ^ ^ _ TlysurKp | A ft^^urrAy^i ^. ty-arp"o wo p -fr^B- turrtn^r 1-6)41 ^ o p-frK «a <u K \jwt» \ trW «-p «j W \jvro •% t- i. t« > U«rt ^ !■ 1 ■ •*•■ JLiyjur- ^ v-tJj ■ 6-a-^apttJ 67 ku lyL-»i &JJ - r • "» ft' ' , ' I ' "" , -j. *v * f-yTj^>»4j« Imly o l T*y^wff1.V «j(yjy-ftwo-, o u tu5^-<n ajjrcoy -retuf f-l*^-*\3V luUtVTttJ t r<r ^ L ^''* J ' f^f^* "^ °^Ui^H rtA(TXop*TT> Uj S-l (X^ftT» ta f^H itr^^V ' ^6t«Ctt) t-ajft-v-fay bdrftfip d^-roy £lr'li_Mty-e". iVtw^tF ^E-^ t^ 4 t-2;' ■an* / t 1 j» &rf4« fr bK^i^oji lufrvrpo'y (^t t-»« rAM tujt.jun/ . ior*n^lp Irvtr"rlni I I I i I 1 1 1 1 _t THE THIRTEEN BOOKS OF EUCLID'S ELEMENTS TRANSLATED FROM THE TEXT OF HEIBERG WITH INTRODUCTION AND COMMENTARY BY Sir THOMAS L. HEATH, K.C.B., K.C.V.O., F.R.S., SC.D. CAMS,, HON. D.SC. OXFORD HONORARY FELLOW (SOMETIME FELLOW) OF TRINITY COLLEGE CAMBRIDGE SECOND EDITION REVISED WITH ADDITIONS VOLUME I INTRODUCTION AND BOOKS I, H DOVER PUBLICATIONS, INC. NEW YORK Published in Canada by General Publishing Com- pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC 2. This Dover edition, first published in 1956, is an unabridged and unaltered republication of the second edition. It is published through special ar- rangement with Cambridge University Press. Library of Congress Catalog Card Number: 56-4H6 Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N.Y. 10014 PREFACE " ' I 'HERE never has been, and till we see it we never J. shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid." De Morgan wrote thus in October 1 848 {Short supplementary remarks on the first six Books of Euclid's Elements in the Companion to the Almanac for 1 849) ; and I do not think that, if he had been living to-day, he would have seen reason to revise the opinion so deliberately pronounced sixty years ago. It is true that in the interval much valuable work has been done on the continent in the investigation of the first principles, including the formulation and classification of axioms or postulates which are necessary to make good the deficiencies of Euclid's own explicit postulates and axioms and to justify the further assumptions which he tacitly makes in certain propositions, content apparently to let their truth be inferred from observa- tion of the figures as drawn ; but, once the first principles are disposed of, the body of doctrine contained in the recent text- books of elementary geometry does not, and from the nature of the case cannot, show any substantial differences from that set forth in the Elements. In England it would seem that far less of scientific value has been done ; the efforts of a multitude of writers have rather been directed towards producing alter- natives for Euclid which shall be more suitable, that is to say, easier, for schoolboys. It is of course not surprising that, in vi PREFACE these days of short cuts, there should have arisen a movement to get rid of Euclid and to substitute a "royal road to geometry " ; the marvel is that a book which was not written for schoolboys but for grown men (as all internal evidence shows, and in particular the essentially theoretical character of the work and its aloofness from anything of the nature of "practical" geometry) should have held its own as a school- book for so long. And now that Euclid's proofs and arrange- ment are no longer required from candidates at examinations there has been a rush of competitors anxious to be first in the field with a new text- book on the more "practical" lines which now find so much favour. The natural desire of each teacher who writes such a text -book is to give prominence to some special nostrum which he has found successful with pupils. One result is, too often, a loss of a due sense of proportion ; and, in any case, it is inevitable that there should be great diversity of treatment. It was with reference to such a danger that Lardner wrote in 1846 : "Euclid once superseded, every teacher would esteem his own work the best, and every school would have its own class book. All that rigour and exactitude which have so long excited the admiration of men of science would be at an end. These very words would lose all definite meaning. Every school would have a different standard; matter of assumption in one being matter of demonstration in another ; until, at length, Geometry, in the ancient sense of the word, would be altogether frittered away or be only considered as a particular application of Arithmetic and Algebra." It is, perhaps, too early yet to prophesy what will be the ultimate outcome of the new order of things ; but it would at least seem possible that history will repeat itself and that, when chaos has come again in geometrical teaching, there will be a return to Euclid more or less complete for the purpose of standardising it once more. But the case for a new edition of Euclid is independent of any controversies as to how geometry shall be taught to schoolboys. Euclid's work will live long after all the text-books PREFACE vii of the present day are superseded and forgotten. It is one of the noblest monuments of antiquity ; no mathematician worthy of the name can afford not to know Euclid, the real Euclid as distinct from any revised or rewritten versions which will serve for schoolboys or engineers. And, to know Euclid, it is necessary to know his language, and, so far as it can be traced, the history of the *' elements " which he collected in his immortal work. This brings me to the raison d'itre of the present edition. A new translation from the Greek was necessary for two reasons. First, though some time has elapsed since the appearance of Heiberg's definitive text and prolegomena, published between 1883 and 1888, there has not been, so far as I know, any attempt to make a faithful translation from it into English even of the Books which are commonly read. And, secondly, the other Books, vn. to X. and xin., were not included by Simson and the editors who followed him, or apparently in any English translation since Williamson's (178 1 — 8), so that they are now practically inaccessible to English readers in any form. In the matter of notes, the edition of the first six Books in Greek and Latin with notes by Camerer and Ha'uber (Berlin, 1824 — 5) is a perfect mine of information. It would have been practically impossible to make the notes more exhaustive at the time when they were written. But the researches of the last thirty or forty years into the history of mathematics (I need only mention such names as those of Bretschneider, Hankel, Moritz Cantor, Hultsch, Paul Tannery, Zeuthen, Loria, and Heiberg) have put the whole subject upon a different plane. I have endeavoured in this edition to take account of all the main results of these researches up to the present date. Thus, so far as the geometrical Books are concerned, my notes are intended to form a sort of dictionary of the history of elementary geometry, arranged according to subjects ; while the notes on the arithmetical Books vii. — ix. and on Book x. follow the same plan. viii PREFACE I desire to express here my thanks to my brother, Dr R. S. Heath, Vice- Principal of Birmingham University, for suggestions on the proof sheets and, in particular, for the reference to the parallelism between Euclid's definition of proportion and Dedekind's theory of irrationals, to Mr R. D. Hicks for advice on a number of difficult points of translation, to Professor A. A. Bevan for help in the transliteration of Arabic names, and to the Curators and Librarian of the Bodleian Library for permission to reproduce, as frontispiece, a page from the famous Bodleian MS, of the Elements. Lastly, my best acknowledgments are due to the Syndics of the Cambridge University Press for their ready acceptance of the work, and for the zealous and efficient cooperation of their staff which has much lightened the labour of seeing the book through the Press. T. L H. Novem$er t 1908. PREFACE TO THE SECOND EDITION I LIKE to think that the exhaustion of the first edition of this work furnishes a new proof (if such were needed) that Euclid is far from being defunct or even dormant, and that, so long as mathematics is studied, mathematicians will find it necessary and worth while to come back again and again, for one purpose or another, to the twenty-two-centuries- old book which, notwithstanding its imperfections, remains the greatest elementary textbook in mathematics that the world is privileged to possess. The present edition has been carefully revised throughout, and a number of passages (sometimes whole pages) have been rewritten, with a view to bringing it up to date. Some not in- considerable additions have also been made, especially in the Excursuses to Volume I, which will, I hope, find interested readers. Since the date of the first edition little has happened in the domain of geometrical teaching which needs to be chronicled. Two distinct movements however call for notice. The first is a movement having for its object the mitigation of the difficulties (affecting in different ways students, teachers and examiners) which are found to arise from the multiplicity of the different textbooks and varying systems now in use for the teaching of elementary geometry. These difficulties have evoked a widespread desire among teachers for the establish- ment of an agreed sequence to be generally adopted in teaching the subject. One proposal to this end has already been made: but the chance of the acceptance of an agreed sequence has in the meantime been prejudiced by a second movement which has arisen in other quarters. x PREFACE TO THE SECOND EDITION I refer to the movement in favour of reviving, in a modified form, the proposal made by Wallis in 1663 to replace Euclid's Parallel- Postulate by a Postulate of Similarity (as to which see pp. 2 10 — 1 1 of Volume I of this work). The form of Postulate now suggested is an assumption that "Given one triangle, there can be constructed, on any arbitrary base, another triangle equiangular with (or similar to) the given triangle." It may perhaps be held that this assumption has the advantage of not referring, in the statement of it, to the fact that a straight line is of unlimited length ; but, on the other hand, as is well known, Saccheri showed ( 1 733) that it involves more than is necessary to enable Euclid's Postulate to be proved. In any case it would seem certain that a scheme based upon the proposed Postulate, if made scientifically sound, must be more difficult than the procedure now generally followed. This being so, and having regard to the facts ( 1 ) that the difference between the suggested Postulate and that of Euclid is in effect so slight and (2) that the historic interest of Euclid's Postulate is so great, I am of opinion that the proposal is very much to be deprecated. T. L. H. December 1925. CONTENTS VOLUME I. INTRODUCTION. Facsimile ofapageofthe Bodleian ms. of the Elements . Frontispiece This is a facsimile of a page (fol. 45 verso) of the famous Bodleian Ms. of the Element i, D'Orville 301 {formerly x. r inf. 1, 30), written in the year 888. The scholium in the margin, not very difficult to decipher, though some letters are almost rubbed out, is one of the scholia Vatican a given by Heiberg (Vol. v. p. 163} as ill. No- 15 : Aid tw xirrpov tOcQv q&k rj v f»rn$«tft aftav, tl Six* Tifiwauti* dWiJXaT * t4 y&p icfwrpor a&r&r 1j Sixrrofdn, bjLoltits xai i} et r$F tripe* Bt& rov Kfrrpov aCenjt 17 trip* ^ 5ta rav KfvTpov tlTj, Sri 06 3fva r4fit*erai 1j && tou Ktvrpoy. The rt before U in the last sentence should be omitted, PFVat. read ij without el. The marginal references lower down are of course to propositions quoted, (1) Sti t& a' rod y. " by III. i," and (aj &i ri 7' toC atow, u by 3 of the same/ 1 Chap. I. II. III. IV. V. VI. VII. VIII. IX. Euclid and the traditions about him Euclid's other works Greek commentators other than Proclus Proclus and his sources .... The Text The Scholia Euclid in Arabia ...... Principal translations and editions . § 1. On the nature of Elements g a. Elements anterior to Euclid's . § 3. First principles : Definitions, Postulates and Axioms ..... Theorems and Problems The formal divisions of a proposition Other technical terms The definitions §4- §5- §7- THE ELEMENTS. Definitions, Postulates, Common Notions Notes on Definitions etc. . Propositions .... Definitions .... Note on geometrical algebra Propositions .... Pythagoras and the Pythagoreans II. Popular names for Euclidean Propositions Greek Index to Vol. I English Index to Vol. I Book I. Book II. Excursus Excursus pack 1 7 19 *9 46 64 75 9i 114 116 117 124 129 132 143 1 S3 i55 241 370 37» 375 411 415 419 421 INTRODUCTION. CHAPTER I. EUCLID AND THE TRADITIONS ABOUT HIM. As in the case of the other great mathematicians of Greece, so in Euclid's case, we have only the most meagre particulars of the life and personality of the man. Most of what we have is contained in the passage of Proclus' summary relating to him, which is as follows 1 : "Not much younger than these (sc. Hermotimus of Colophon and Philippus of Med ma) is Euclid, who put together the Elements, collect- ing many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived* in the time of the first Ptolemy. For Archimedes, who came imme- diately after the first (Ptolemy)', makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry*. He is then younger than the pupils of Plato but older than Eratosthenes and Archimedes ; for the latter were contemporary with one another, as Eratosthenes some- where says." This passage shows that even Proclus had no direct knowledge of Euclid's birthplace or of the date of his birth or death. He pro- ceeds by inference. Since Archimedes lived just after the first 1 Proclus, ed. Fried lein, p. 68, 6 — io, * The word ytyott must apparently mean " nourished," as Heiberg understands it (LitttrttrgathicktHeht Stttditn itber EtMid, 188,1, p. 16), not "was born," as Hankel took it ; otherwise part of Proclus' argument would lose its cogency. * So Heiberg understands ?n/iii\£iv n? *p&Ti# (sc. IlToXe^ttV). Friedlein's text has vol between Iwifi&Xuv And t<£ *p&ry \ and it is right to remark that another reading is ttX it Tif Trptirif (without trtpaXiiy'i which has been translated " in his first book," by which is understood On the Sphere and Cylinder I., where (i) in Prop- 3 »™ the words " let BC be made equal to D by tkt second (proposition) of the ftrit of Euclid's (books)," and (i) in Prop, fi the words " For these things are handed down in the Elements " (without the name of Euclid). Heiberg thinks the former passage is referred to, and that Proclus must therefore have had before him the words " by the second of the first of Euclid ": a fair proof that they are genuine, though in themselves they would be somewhat suspicious. 4 The same story is told in Stobaeus, Eel. (it. p. m8, 30, ed, Wachsmuthf about Alexander and Menaechmus. Alexander is represented as having asked Menaechmus to teach him geometry concisely, but he replied 1 "O king, through the country there are royal roads and road* for common citizen*, but in geometry there is one road for all." * INTRODUCTION [ch. i Ptolemy, and Archimedes mentions Euclid, while there is an anecdote about some Ptolemy and Euclid, therefore Euclid lived in the time of the first Ptolemy. We may infer then from Proclus that Euclid was intermediate between the first pupils of Plato and Archimedes. Now Plato died in 347/6, Archimedes lived 287-2 1 2, Eratosthenes c. 284-204 B.C. Thus Euclid must have flourished c. 300 b.c, which date agrees well with the fact that Ptolemy reigned from 306 to 283 B.C. It is most probable that Euclid received his mathematical training in Athens from the pupils of Plato; for most of the geometers who could have taught him were of that school, and it was in Athens that the older writers of elements, and the other mathematicians on whose works Euclid's Elements depend, had lived and taught. He may himself have been a Platonist, but this does not follow from the state- ments of Proclus on the subject, Proclus says namely that he was of the school of Plato and in close touch with that philosophy 1 . But this was only an attempt of a New Platonist to connect Euclid with his philosophy, as is clear from the next words in the same sentence, " for which reason also he set before himself, as the end of the whole Elements, the construction of the so-called Platonic figures." It is evident that it was only an idea of Proclus' own to infer that Euclid was a Platonist because his Elements end with the investigation of the five regular solids, since a later passage shows him hard put to it to reconcile the view that the construction of the five regular solids was the end and aim of the Elements with the obvious fact that they were intended to supply a foundation for the study of geometry in general, "to make perfect the understanding of the learner in regard to the whole of geometry*." To get out of the difficulty he says* that, if one should ask him what was the aim {o-kowos) of the treatise, he would reply by making a distinction between Euclid's intentions ( 1) as regards the subjects with which his investigations are concerned, (2) as regards the learner, and would say as regards ( 1 ) that " the whole of the geometer's argument is concerned with the cosmic figures." This latter statement is obviously incorrect It is true that Euclid's Elements end with the construction of the five regular solids ; but the pi an i metrical portion has no direct relation to them, and the arithmetical no relation at all ; the propositions about them are merely the conclusion of the stereo metrical division of the work. One thing is however certain, namely that Euclid taught, and founded a school, at Alexandria. This is clear from the remark of Pappus about Apollonius*: "he spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought," It is in the same passage that Pappus makes a remark which might, to an unwary reader, seem to throw some light on the 1 Proclus, p. 68, JO, ul tJ rpoatpim ti lUaTwiKii itrn cal r^i 0i\<wo0Ja twJtj eJituw. 1 i&jd, p. 7 [, 8. * ibid. p. 70, 19 jqq. * Pappus, VII, p, 678, 10 — II, CVffxokAjrat rat i>*& EvxXd&ov jukthjrait tr 'AX*(aj>?pr£p ch. i] EUCLID AND TRADITIONS ABOUT HIM 3 personality of Euclid. He is speaking about Apollonius' preface to the first book of his Conies, where he says that Euclid had not completely worked out the synthesis of the " three- and four-line locus," which in fact was not possible without some theorems first discovered by himself. Pappus says on this 1 : "Now Euclid — regarding Aristaeus as deserving credit for the discoveries he had already made in conies, and without anticipating him or wishing to construct anew the same system (such was his scrupulous fairness and his exemplary kindliness towards all who could advance mathematical science to however small an extent), being moreover in no wise con- tentious and, though exact, yet no braggart like the other [Apollonius] — wrote so much about the locus as was possible by means of the conies of Aristaeus, without claiming completeness for his demonstra- tions." It is however evident, when the passage is examined in its context, that Pappus is not following any tradition in giving this account of Euclid : he was offended by the terms of Apollonius' reference to Euclid, which seemed to him unjust, and he drew a fancy picture of Euclid in order to show Apollonius in a relatively unfavourable light. Another story is told of Euclid which one would like to believe true. According to Stobaeus*, "some one who had begun to read geometry with Euclid, when he had learnt the first theorem, asked Euclid, ' But what shall I get bylearning these things?' Euclid called his slave and said 'Give him threepence, since he must make gain out of what he learns.' " In the middle ages most translators and editors spoke of Euclid as Euclid of Megara. This description arose out of a confusion between our Euclid and the philosopher Euclid of Megara who lived about 400 B.C. The first trace of this confusion appears in Valerius Maximus (in the time of Tiberius) who says* that Plato, on being appealed to for a solution of the problem of doubling the cubical aitar, sent the inquirers to " Euclid the geometer." There is no doubt about the reading, although an early commentator on Valerius Maximus wanted to correct " Eucliden " into " Eudoxum? and this correction is clearly right. But, if Valerius Maximus took Euclid the geometer for a contemporary of Plato, it could only be through confusing him with Euclid of Megara. The first specific reference to Euclid as Euclid of Megara belongs to the 14th century, occurring in the v^rofivtifuiTitrfial of Theodorus Metochita (d. 1332) who speaks of " Euclid of Megara, the Socratic philosopher, contemporary of Plato," as the author of treatises on plane and solid geometry, data, optics etc. : and a Paris MS. of the 14th century has " Euctidis philosophi Socratici liber ele mentor urn," The misunderstanding was general in the period from Campanus 1 translation (Venice 1482) to those of Tartaglia (Venice 1565) and Candalla (Paris 1566). But one Constantinus Lascaris (d. about 1493) had already made the proper 1 Pappus, VII. pp. €76, 55 — 678* 6. Hultsch, it is true, brackets the whole passage ppu 676, i j— 678, I j, but apparently on the ground of the diction only, * Stobaeus, I.e. * vill. n, eit. i. 4 INTRODUCTION [ch. i distinction by saying of our Euclid that "he was different from him of Megara of whom Laertius wrote, and who wrote dialogues "' ; and to Commandinus belongs the credit of being the first translator* to put the matter beyond doubt : " Let us then free a number of people from the error by which they have been induced to believe that our Euclid is the same as the philosopher of Megara " etc Another idea, that Euclid was born at Gela in Sicily, is due to tne same confusion, being based on Diogenes Laertius* description' of the philosopher Euclid as being "of Megara, or, according to some, of Gela, as Alexander says in the AiaBoxai-" In view of the poverty of Greek tradition on the subject even as early as the time of Proclus (410-485 A.D.), we must necessarily take cum gram the apparently circumstantial accounts of Euclid given by Arabian authors ; and indeed the origin of their stories can be explained as the result (1) of the Arabian tendency to romance, and (2) of misunderstandings. We read* that " Euclid, son of Naucrates, grandson of Zenarchus", called the author of geometry, a philosopher of somewhat ancient date, a Greek by nationality domiciled at Damascus, born at Tyre, most learned in the science of geometry, published a most excellent and most useful work entitled the foundation or elements of geometry, a subject in which no more general treatise existed before among the Greeks : nay, there was no one even of later date who did not walk in his footsteps and frankly profess his doctrine. Hence also Greek, Roman and Arabian geometers not a few, who undertook the task of illustrating this work, published commentaries, scholia, and notes upon it, and made an abridgment of the work itself. For this reason the Greek philosophers used to post up on the doors of their schools the well-known notice : ' Let no one come to our school, who has not first learned the elements of Euclid.' " The details at the beginning of this extract cannot be derived from Greek sources, for even Proclus did not know anything about Euclid's father, while it was not the Greek habit to record the names of grandfathers, as the Arabians commonly did. Damascus and Tyre were no doubt brought in to gratify a desire which the Arabians always showed to connect famous Greeks in some way or other with the East. Thus Nasiraddin, the translator of the Elements, who was of Tus in Khurasan, actually makes Euclid out to have been " Thusinus " also*. The readiness of the Arabians to run away with an idea is illustrated by the last words 1 Letter to Fernandus Acuna, printed in Maurolycus, Hitteria Sitiliat, fol. 11 r. (see Heiberg, EuMid-Studieriy pp. n — 3, 35). ■ Preface to translation (Pisauri, 1571). * Diog. L. 11. 106, p. ;8 ed. Cobet. * Casiri, Bibliotheca Arabifo-Hispana Escuriaiensis, 1. p. 339, Casirt's source is al- Qifti (d. 1 148). trie author of the Ta'rttk ai-ffu&amd, a collection of biographies of phi- losophers, mathematicians, astronomers etc ■ The Fi'Aris/ says "son of Naucrates, the son of Berenice f/) " (see Suter"s translation in Abhandiungtn tur Gach. d. Math. V], Heft, 1H91. p. 16). e The same predilection made the Arabs describe 'Pythagoras as a pupil of the wise Salomo, Hippftrcbus as the exponent of Chaldaean philosophy or as the Chaldaean, Archi- medes as an Egyptian etc. (Hij! Khalfa, Lexicon BiMiogrspkitvm, and Casiri). ch. i) EUCLID AND TRADITIONS ABOUT HIM 5 of the extract. Everyone knows the story of Plato's inscription over the porch of the Academy : " let no one unversed in geometry enter my doors " ; the Arab turned geometry into Euclid 's geometry, and told the story of Greek philosophers in general and ''their Academies," Equally remarkable are the Arabian accounts of the relation of Euclid and Apollonius 1 . According to them the Elements were originally written, not by Euclid, but by a man whose name was Apollonius, a carpenter, who wrote the work in 1 5 books or sections'. In the course of time some of the work was lost and the rest became disarranged, so that one of the kings at Alexandria who desired to study geometry and to master this treatise in particular first questioned about it certain learned men who visited him and then sent for Euclid who was at that time famous as a geometer, and asked him to revise and complete the work and reduce it to order. Euclid then re- wrote h in 13 books which were thereafter known by his name. (According to another version Euclid composed the 1 3 books out of commentaries which he had published on two books of Apollonius on conies and out of introductory matter added to the doctrine of the five regular solids.) To the thirteen books were added two more books, the work of others (though some attribute these also to Euclid) which contain several things not mentioned by Apollonius. According to another version Hypsicles, a pupil of Euclid at Alexandria, offered to the king and published Books xiv. and xv., it being also stated that Hypsicles had " discovered " the books, by which it appears to be suggested that Hypsicles had edited them from materials left by Euclid. We observe here the correct statement that Books XIV. and xv. were not written by Euclid, but along with it the incorrect informa- tion that Hypsicles, the author of Book XIV., wrote Book XV. also. The whole of the fable about Apollonius having preceded Euclid and having written the Elements appears to have been evolved out of the preface to Book xiv. by Hypsicles, and in this way ; the Book must in early times have been attributed to Euclid, and the inference based upon this assumption was left uncorrected afterwards when it was recognised that Hypsicles was the author. The preface is worth quoting : "Basiiides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of their common interest in mathematics. And once, when 1 The authorities for these statements quoted by Casiri and lliji Khalfa are al-Kindl '3 tract de inttituto tibri Eutlidis (al-Kindl died about 873) and a commentary by Qidliide ar-Ruml (<[. about 1440) on a book called Askk&l ai-ttf sis (fundamental propositions) by Ashraf Shanuaddln aS'Samaru&ndi (c. 1176) consisting of elucidations of 3$ propositions ■elected from the first books of Euclid. Naslraddin likewise says that Euclid cut out two of 15 books of elements then existing and published the rest under his own name. According to Qadtzade the king heard that there was a celebrated geometer named Euclid at Tyrt\ Naslr- atldin says that he sent for Euclid of Tus. * So says the Fihrist, Suter {<>p- cit, p. 49) thinks that the author of the Fihrist did not suppose Apollonius of Ptrga to be the writer of the Ettmmts, as later Arabian authorities did, but that he disUnguished another Apollonius whom he calls " a carpenter." Suter's argument is based on the fact that the Fihrist s article on Apollonius (of Perga) says nothing of the Ebmttus, aiiJ that it gives the three great mathematicians, Euclid, Archimedes and Apollonius, in the correct chronological order. 6 INTRODUCTION [ch. i examining the treatise written by Apollonius about the comparison between the dodecahedron and the icosahedron inscribed in the same sphere, (showing) what ratio they have to one another, they thought that Apollonius had not expounded this matter properly, and accordingly they emended the exposition, as I was able to learn from my father. And I myself, later, fell in with another book published by Apollonius, containing a demonstration relating to the subject, and I was greatly interested in the investigation of the problem. The book published by Apollonius is accessible to all— for it has a large circulation, having apparently been carefully written out later — but I decided to send you the comments which seem to me to be necessary, for you will through your proficiency in mathe- matics in general and in geometry in particular form an expert judgment on what I am about to say, and you will lend a kindly ear to my disquisition for the sake of your friendship to my father and your goodwil! to me." The idea that Apollonius preceded Euclid must evidently have been derived from the passage just quoted. It explains other things besides. Basil ides must have been confused with j3a<7t\ev$, and we have a probable explanation of the " Alexandrian king," and of the " learned men who visited " Alexandria. It is possible also that in the " Tyrian " of Hypsicles" preface we have the origin of the notion that Euclid was born in Tyre. These inferences argue, no doubt, very defective knowledge of Greek : but we could expect no better from those who took the Organon of Aristotle to be " instrumentum musicum pneumaticum," and who explained the name of Euclid, which they variously pronounced as U elides or /eludes, to be com- pounded of Ueli a key, and Dis a measure, or, as some say, geometry, so that [/elides is equivalent to the key of geometry '! Lastly the alternative version, given in brackets above, which says that Euclid made the Elements out of commentaries which he wrote on two books of Apollonius on conies and prolegomena added to the doctrine of the five solids, seems to have arisen, through a like confusion, out of a later passage 1 in Hypsicles' Book XIV. : " And this is expounded by Aristaeus in the book entitled 'Comparison of the five figures,' and by Apollonius in the second edition of his comparison of the dodecahedron with the icosahedron." The "doctrine of the five solids" in the Arabic must be the "Comparison of the five figures" in the passage of Hypsicles, for nowhere else have we any information about a work bearing this title, nor can the Arabians have had. The reference to the two books of Apollonius on conies will then be the result of mixing up the fact that Apollonius wrote a book on conies with the second edition of the other work mentioned by Hypsicles. We do not find elsewhere in Arabian authors any mention of a commentary by Euclid on Apollonius and Aristaeus: so that the story in the passage quoted is really no more than a variation of the fable that the Elements were the work of Apollonius. 1 Heibtrg*5 Euclid, vol. v. p. 6. CHAPTER II. EUCLID'S OTHER WORKS. In giving a list of the Euclidean treatises other than the Elements, I shall be brief: for fuller accounts of them, or speculations with regard to thern, reference should be made to the standard histories of mathematics 1 . I will take first the works which are mentioned by Greek authors, i. The Pseudaria. I mention this first because Proclus refers to it in the general remarks in praise of the Elements which he gives immediately after the mention of Euclid in his summary. He says'; " But, inasmuch as many things, while appearing to rest on truth and to follow from scientific principles, really tend to lead one astray from the principles and deceive the more superficial minds, he has handed down methods for the discriminative understanding of these things as well, by the use of which methods we shall be able to give beginners in this study practice in the discovery of paralogisms, and to avoid being misled. This treatise, by which he puts this machinery in our hands, he entitled (the book) of Pseudaria, enumerating in order their various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of error with practical illustration. This book then is by way of cathartic and exercise, while the Elements contain the irrefragable and complete guide to the actual scientific investigation of the subjects of geometry." The book is considered to be irreparably lost. We may conclude however from the connexion of it with the Elements and the reference to its usefulness for beginners that it did not go outside the domain of elementary geometry'. 1 See, for example, Lena, Le seiente esatte neW antica Grata, 1914, pp. 145- — 168; T. L. Heath, History of Greek Mathematics, 191 1, J. pp. \i\ — 446. Cf- Heiberg, Litttrar- gatkUhtliche Studim tibar Euklid, pp. 36 — 153; Euclidis opera omnia, ed. Heiberg and Menge, Vols. VI. — VLI1, ■ Proclus, p. 70, i — 18. 3 Heiberg points out that Alexander Aphrodisiensis appears to allude to the work in his commentary on Aristotle's Sophistiti Ehncki {fol. 25 b) : "Not only those {fheyx *) which do not start from the principles of the science under which the problem is classed... but also those which do start from the proper principles of the science but in some respect admit a paralogism, e.g. the Ptevobgraphemaia of Euclid." Tannery {Butt, ties jeiencei ma/A- et aifr. »• Serie, vi., 1881, [*" Partie, p. 147) conjectures that it may be from this treatise that the : commentator got his Information about the quadratures of the circle by Antiphon and 8 INTRODUCTION [ch. n 2. The Data. The Data (Se&oftiva) are included by Pappus in the Treasury of Analysis (x6tto<; avrikvo/ievoi;), and he describes their contents 1 . They are still concerned with elementary geometry, though forming part of the introduction to higher analysis. Their form is that of pro- positions proving that, if certain things in a figure are given (in magnitude, in species, etc.), something else is given. The subject- matter is much the same as that of the planimetrical books of the Elements, to which the Data are often supplementary. We shall see this later when we come to compare the propositions in the Elements which give us the means of solving the general quadratic equation with the corresponding propositions of the Data which give the solution. The Data may in fact be regarded as elementary exercises in analysis. It is not necessary to go more closely into the contents, as we have the full Greek text and the commentary by Marinus newly edited by Menge and therefore easily accessible 1 . 3. The book On divisions {of figures). This work (irepl Statpetre&v fiiffklov) is mentioned by Proclus*. In one place he is speaking of the conception or definition (\6yo<t) of figure, and of the divisibility of a figure into others differing from it in kind ; and he adds : " For the circle is divisible into parts unlike in definition or notion (eivo/ioia rji Xtfyy), and so is each of the rectilineal figures ; this is in fact the business of the writer of the Elements in his Divisions, where he divides given figures, in one case into like figures, and in another into unlike*." "Like" and "unlike" here mean, not "similar" and "dissimilar" in the technical sense, but "like" or "unlike in definition or notion" (\6y<p): thus to divide a triangle into triangles would be to divide it into "like" figures, to divide a triangle into a triangle and a quadrilateral would be to divide it into "unlike'' figures. The treatise is lost in Greek but has been discovered in the Arabic. First John Dee discovered a treatise De divisionibus by one Muhammad Bagdadinus* and handed over a copy of it (in Latin) in 1563 to Commandinus. who published it, in Dee's name and his own, in 1570 s , Dee did not himself translate the tract from the Arabic; he Brjrccm, to say nothing of the lunules of Hippocrates. I think however that there is is objection to this theory so far as regards Bryson; for Alexander distinctly says that Bryson'i quadrature did not start from the proper principles of geometry, but from some principles more general. 1 Pappus, vn. p. 638. * Vol. vi. in the Teubner edition of Euclidis Optra omnia by Heiberg and Menge. A translation of the Data is also included in Simson*3 Euclid (though naturally his text left much to be desired). 8 Proclus, p. 69, 4. * ibid. 144, Ji— 26. * Stetnschneider places him in the roth c. H. Suter (BiUiothtea Mathematics, 1v„ 1903, pp. «4, 17} identifies him with Abu (Bekr) Muh. b. 'Abdalbaq! al-Bagdadl, QadI (Judge) of Maristan (circa rt>70-i 14 1), to whom he also attributes the Likcrjudci (? judicis) super detimum Euciidis translated by Gherard of Cremona. ' Dc atperficicrum divisionibui liber Mackomcto Bagdadino adscripUa, nunc primam /oannis Da Londitunsis tt Federici Ctsmmandini Urbinatti opera in lutein cditus, Pisauri, 1570, afterwards included in Gregory's Euclid (Oxford, r7oj). ch. n] EUCLID'S OTHER WORKS 9 found it in Latin in a ms, which was then in his own possession but was about 20 years afterwards stolen or destroyed in an attack by a mob on his house at Mortlake'. Dee, in his preface addressed to Commandinus, says nothing of his having translated the book, but only remarks that the very illegible ms. had caused him much trouble and (in a later passage) speaks of " the actual, very ancient, copy from which I wrote out..." (in ipso unde descripsi vetustissimo exemplari). The Latin translation of this tract from the Arabic was probably made by Gherard of Cremona (1 1 14- 1 1 87), among the list of whose numerous translations a " liber divisionum " occurs. The Arabic original cannot have been a direct translation from Euclid, and probably was not even a direct adaptation of it ; it contains mistakes and un mathematical expressions, and moreover does not contain the propositions about the division of a circle alluded to by Proclus. Hence it can scarcely have contained more than a fragment of Euclid's work. But Woepcke found in a MS. at Paris a treatise in Arabic on the division of figures, which he translated and published in 1851*. It is expressly attributed to Euclid in the MS. and corresponds to the description of it by Proclus. Generally speaking, the divisions are divisions into figures of the same kind as the original figures, e.g. of triangles into triangles ; but there are also divisions into " unlike " figures, e.g. that of a triangle by a straight line parallel to the base. The missing propositions about the division of a circle are also here: " to divide into two equal parts a given figure bounded by an arc of a circle and two straight lines including a given angle " and " to draw in a given circle two parallel straight lines cutting off a certain part of the circle." Unfortunately the proofs are given of only four propositions (including the two last mentioned) out of 36, because the Arabic translator found them too easy and omitted them. To illustrate the character of the problems dealt with I need only take one more example : " To cut off a certain fraction from a (parallel-) trapezium by a straight line which passes through a given point lying inside or outside the trapezium but so that a straight line can be drawn through it cutting both the parallel sides of the trapezium," The genuineness of the treatise edited by Woepcke is attested by the facts that the four proofs which remain are elegant and depend on propositions in the Elements, and that there is a lemma with a true Greek ring: "to apply to a straight line a rectangle equal to the rectangle contained by AB, AC and deficient by a square." Moreover the treatise is no fragment, but finishes with the words "end of the treatise," and is a well-ordered and compact whole. Hence we may safely conclude that Woepcke's is not only Euclid's own work but the whole of it. A restoration of the work, with proofs, was attempted by Ofterdinger 3 , who however does not give Woepcke's props. 30, 31, 34. 35. 36- VVe have now a satisfactory restoration, with ample notes 1 R. C. Archibald, Euclid's Boot: on the Division of Figures with a restoration based on Wotpehts text and on the Praetita geometriae of Leonardo Pisano^ Cambridge, 19(5, pp. 4 — 9. ' Journal A natiqut, |8JI, p. 133 sqq. 3 L. F. Ofterdinger, Beitrtige tur Wiederherslettung dcr Sehrifi dot Eu&lides iibrr die Tkeilmtg der Figurtn, Ulm, 1BJ3. io INTRODUCTION [ch. n and an introduction, by R. C. Archibald, who used for the purpose Woepcke's text and a section of Leonardo of Pisa's Practica geometriae (1220) 1 . 4. The Porisms. It is not possible to give in this place any account of the con- troversies about the contents and significance of the three lost books of Porisms, or of the important attempts by Robert Simson and Chasles to restore the work. These may be said to form a whole literature, references to which will be found most abundantly given by Heiberg and Loria, the former of whom has treated the subject from the philological point of view, most exhaustively, while the latter, founding himself generally on Heiberg, has added useful details, from the mathematical side, relating to the attempted restora- tions, etc' It must suffice here to give an extract from the only original source of information about the nature and contents of the Porisms, namely Pappus*. In his general preface about the books composing the Treasury of Analysis (towot dvaXuofttvos) he says : "After the Tan gene ies (of Apollonius) come, in three books, the Porisms of Euclid, [in the view of many] a collection most ingeniously devised for the analysis of the more weighty problems, [and] although nature presents an unlimited number of such porisms' 1 , [they have added nothing to what was written originally by Euclid, except that some before my time have shown their want of taste by adding to a few (of the propositions) second proofs, each (proposition) admitting of a definite number of demonstrations, as we have shown, and Euclid having given one for each, namely that which is the most lucid. These porisms embody a theory subtle, natural, necessary, and of considerable generality, which is fascinating to those who can see and produce results]. " Now all the varieties of porisms belong, neither to theorems nor problems, but to a species occupying a sort of intermediate position [so that their enunciations can be formed like those of either theorems or problems], the result being that, of the great number of geometers, some regarded them as of the class of theorems, and others of pro- blems, looking only to the form of the proposition. But that the ancients knew better the difference between these three things is clear from the definitions. For they said that a theorem is that which is proposed with a view to the demonstration of the very thing proposed, a problem that which is thrown out with a view to the construction of the very thing proposed, and a porism that which is proposed with a view to the producing of the very thing proposed. [But this definition of the porism was changed by the more recent writers who could not produce everything, but used these elements 1 There is a remarkable similarity between the propositions of Wot octet's text and those of Leonardo, suggesting that Leonardo may have had before him a translation (perhaps by Gherard of Cremona) of the Arabic tract. * Heiberg, liu&lid-Studiin, pp. 56 — 79, and Loria, op. at., pp. 1 S3 — jfij. ' Pappus, ed. Hullsch, VII. pp. 648 — 660. I put in square brackets the words bracketed by Haltsch. ' I adopt Heiberg 1 s reading of a comma here instead of a full stop. ch. it] EUCLID'S OTHER WORKS n and proved only the fact that that which is sought really exists, but did not produce it 1 and were accordingly confuted by the definition and the whole doctrine. They based their definition on an incidental characteristic, thus : A porism is that which falls short of a locus- theorem in respect of its hypothesis'. Of this kind of porisms loci are a species, and they abound in the Treasury of Analysis ; but this species has been collected, named and handed down separately from the porisms, because it is more widely diffused than the other species]. But it has further become characteristic of porisms that, owing to their complication, the enunciations are put in a contracted form, much being by usage left to be understood; so that many geometers understand them only in a partial way and are ignorant of the more essential features of their contents, "[Now to comprehend a number of propositions in one enunciation is by no means easy in these porisms, because Euclid himself has not in fact given many of each species, but chosen, for examples, one or a few out of a great multitude*. But at the beginning of the first book he has given some propositions, to the number of ten, of one species, namely that more fruitful species consisting of loci.] Consequently, finding that these admitted of being comprehended in one enunciation, we have set it out thus: If, in a system of four straight lines' which cut each other two and two, three points on one straight line be given while the rest except one lie on different straight lines given in position, the remaining point also will lie on a straight line given in position*. 1 Heiberg points out that Props. 5—9 of Archimedes' treatise On Spirals are porisms in this sense. To take Prop. 5 as an example, DBF is a tangent to a circle with centre K- It is then possible, says Archimedes, to draw a straight line r> b F KffF t meeting the circumference in // and the tangent in F, such that FH;HJC^{*kBH):c, where c is the circumference of any circle. To prove this he assumes the following construction. E being any straight line greater than c, he says 1 let KG be parallel to DF, "and let the line GH equal to E be placed verging to the point B." Archimedes must of course nave known how to effect this construction, which requires conies. But that it is possible requires very little argument, for if we draw any straight line BHG meeting the circle in /if and KG in G, it is obvious that as G moves away from C, HG becomes greater and greater and may be made as great as we please. The " later writers " would no doubt have contented themselves with this considera- tion without actually tanstrtuting HG. 1 As Heiberg says, this translation is made certain by a preceding passage of Pappus (p. 648, 1 — 3) where he compares two enunciations, the latter of which " falls short of the former in hypothesis but goes beyond it in requirement" E.g. the first enunciation requiring us, given three circles, to draw a circle touching all three, the second may require us, given only row circles (one less datum), to draw a circle touching them and ef a given sat (an extra requirement). * I translate Heiberg 's reading with a full stop here followed by r/wi ipxi U tiu*t [rpb* &PXV* ($t&Oneror) Hultsch] TOU TpwTOU /Sl^X/ou,,., 1 The four straight lines are described in the text as (the sides) brriov 1 9 wapuwrtout i.e. aides of two' sotts of quadrilaterals which Simson tries to explain {see p. no of the Index Grtueitatis of Hultsch's edition of Pappus). * In other words (Chasles, p, 13; Loria, p. 356), if a triangle be so deformed that each of its sides turns about one of three points in a straight line, and two of its vertices lie on two straight lines given in position, the third vertex will also lie on a straight line. is INTRODUCTION [ch. n "This has only been enunciated of four straight lines, of which not more than two pass through the same point, but it is not known (to most people) that it is true of any assigned number of straight lines if enunciated thus : If any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, and if each of those which are on another (of them) lie on a straight line given in position — or still more generally thus : if any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, while of the other points of intersection in multitude equal to a triangular number a number corresponding to the side of this triangular number lie respectively on straight lines given in position, provided that of these latter points no three are at the angular points of a triangle («. having for sides three of the given straight lines) — each of the remaining points will He on a straight line given in position'. ■ It is probable that fhe writer of the Elements was not unaware of this but that he only set out the principle ; and he seems, in the case of all the porisms, to have laid down the principles and the seed only [of many important things], the kinds of which should be distinguished according to the differences, not of their hypotheses, but of the results and the things sought [All the hypotheses are different from one another because they are entirely special, but each of the results and things sought, being one and the same, follow from many different hypotheses.] "We must then in the first book distinguish the following kinds of things sought : "At the beginning of the book' is this proposition : I. ' If from two given points straight tines be drawn meeting on a straight line given in position, and one cut off from a straight line given in position (a segment measured} to a given point on it, the other will also cut off from another {straight line a segment) liaving to the first a given ratio' " Following on this (we have to prove) II. that such and such a point lies on a straight line given in position ; III. that the ratio of such and such a pair of straight lines is given ; " etc. etc, (up to xxix.). "The three books of the porisms contain 38 lemmas; of the theorems themselves there are 171." 1 Loria (p. as6, ». 3) gives the meaning of this as follows, pointing out that Sim son vu the discoverer of it : " If a complete n-lateral be deformed so that its sides respectively turn about n points on a straight line, and {» ~ 1) of iEs n (« - i)/i vertices move on as many straight lines, the other (» - [)(»- j)/i of its vertices likewise move on as many straight lines ; but it is necessary that it should be impossible to form with the {ft - 1) vertices any triangle having for sides the sides of the polygon." 1 Reading, with Heiberg, roB pipMov [tuS f Hulischl. ch. n] EUCLID'S OTHER WORKS 13 Pappus further gives lemmas to the Porisms (pp. 866 — 918, ed. Hultsch). With Pappus' account of Forisms must be compared the passages of Proclus on the same subject Proclus distinguishes two senses in which the word ir6pierfia is used. The first is that of corollary where something appears as an incidental result of a proposition, obtained without trouble or special seeking, a sort of bonus which the investi- gation has presented us with 1 . The other sense is that of Euclid's Porisms*, In this sense* "porism is the name given to things which are sought, but need some finding and are neither pure bringing into existence nor simple theoretic argument. For (to prove) that the angles at the base of isosceles triangles are equal is a matter of theoretic argument, and it is with reference to things existing that such knowledge is (obtained). But to bisect an angle, to construct a triangle, to cut off, or to place — all these things demand the making of something ; and to find the centre of a given circle, or to find the greatest common measure of two given commensurable magnitudes, or the like, is in some sort between theorems and problems. For in these cases there is no bringing into existence of the things sought, but finding of them, nor is the procedure purely theoretic. For it is necessary to bring that which is sought into view and exhibit it to the eye. Such are the porisms which Euclid wrote, and arranged in three books of Porisms. Proclus' definition thus agrees well enough with the first, " older," definition of Pappus. A porism occupies a place between a theorem and a problem ; it deals with something already existing, as a theorem does, but has to find it (e.g. the centre of a circle), and, as a certain operation is therefore necessary, it partakes to that extent of the nature of a problem, which requires us to construct or produce some- thing not previously existing. Thus, besides III. 1 of the Elements and X. 3, 4 mentioned by Proclus, the following propositions are real porisms: III. 25, VL 11— 13, vn, 33, 34, 36, 39, vm. 2, 4, x. 10, XIII. 18. Similarly in Archimedes On the Sphere and Cylinder I. 2 — 6 might be called porisms. The enunciation given by Pappus as comprehending ten of Euclid's propositions may not reproduce the form of Euclid's enunciations ; but, comparing the result to be proved, that certain points lie on straight lines given in position, with the class indicated by II. above, where the question is of such and such a point lying on a straight line given in position, and with other classes, e.g. (v.) that such and such a line is given in position, (VI.) that such and such a line verges to a given point, (XXVII.) that there exists a given point such that straight lines drawn from it to such and such (circles) will contain a triangle given in species, we may conclude that a usual form of a porism was " to prove that it is possible to find a point with such and such a property" 1 Produsjpp. 111, 14; 301, 11. ■ ibid. p. ill, 11. "The term porism is used of certain problems, like the Ptrrisms written by Euclid." ' ibid. pp. joi, 15 sqq. i 4 INTRODUCTION [ch. it or "a straight line on which lie all the points satisfying given conditions " etc. Sim son defined a porism thus : " Porisma est propositi o in qua proponitur demonstrate rem aliquant, vel pi u res datas esse, cui, vet quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quae ad ea quae data sunt eandem habent re lat ion em, con venire ostendendum est affectionem quandam communem in propositione descriptam 1 ." From the above it is easy to understand Pappus' statement that loci constitute a large class of porisms. A locus is well defined by Simson thus : " Locus est propositio in qua propositum est datam esse demonstrare, vel invenire lineam aut superficiem cuius quodlibet punctum, vel superficiem in qua quaelibet linea data lege descripta, communem quandam ha bet proprietatem in propositione descriptam," Heiberg cites an excellent instance of a loats which is a porism, namely the following proposition quoted by Eutocius' from the Plane Loci of Apollonius : " Given two points in a plane, and a ratio between unequal straight lines, it b possible to draw, in the plane, a circle such that the straight lines drawn from the given points to meet on the circumference of the circle have (to one another) a ratio the same as the given ratio," A difficult point, however, arises on the passage of Pappus, which says that a porism is " that which, in respect of its hypothesis, falls short of a locus-theorem " (rovacov StrnptftuiTos:). Heiberg explains it by comparing the porism from Apollonius' Plane Loci just given with Pappus' enunciation of the same thing, to the effect that, if from two given points two straight lines be drawn meeting in a point, and these straight lines have to one another a given ratio, the point will lie on either a straight line or a circumference of a circle given in position. Heiberg observes that in this latter enunciation something is taken into the hypothesis which was not in the hypothesis of the enunciation of the porism, viz. " that the ratio of the straight lines is the same." I confess this does not seem to me satisfactory : for there is no real difference between the enunciations, and the supposed difference in hypothesis is very like playing with words. Chasles says : " Ce qui constitue le porisme est ce qui manque a I'hypotklse d'un tkioreme local (en d'autres termes, le porisme est inferieur, par l'hypothese, au th^oreme local; e'est-a-dire que quand quelques parties d'une pro- position locale n'ont pas dans l'enonce la determination qui ieur est propre, cette proposition cesse d'etre regardee comme un th^oreme et devient un porisme)." But the subject still seems to require further elucidation. While there is so much that is obscure, it seems certain (i) that the Porisms were distinctly part of higher geometry and not of elementary 1 This was thus expressed by Chasles : n Le porisme est une proposition dans laquelle on demande de demontrer qu'une chose ou plusieurs choses sont denrUts, qui, ainsi que Tone 3uelconque d'une infinite d'autres choses non donnees, jnais dont chacune est avec des choses onntes dans une meme relation, ont une certaine propnete commune, decrite dans 1ft pro- position." * Commentary on Apollonius" Conies (vol. tt. p. ISO, ed. Heiberg). ch. ii] EUCLID'S OTHER WORKS 15 geometry, (a) that they contained propositions belonging to the modern theory of transversals and to projective geometry. It should be remembered too that it was in the course of his researches on this subject that Chasles was led to the idea of anharmonk ratios. Lastly, allusion should be made to the theory of Zeuthen 1 on the subject of the porisms. He observes that the only porjsm of which Pappus gives the complete enunciation, " If from two given points straight lines be drawn meeting on a straight line given in position, and one cut off from a straight line given in position (a segment measured) towards a given point on it, the other will also cut off from another (straight line a segment) bearing to the first a given ratio," is also true if there be substituted for the first given straight line a conic regarded as the "locus with respect to four lines," and that this extended porism can be used for completing Apollonius' exposition of that locus. Zeuthen concludes that the Porisms were in part by- products of the theory of conies and in part auxiliary means for the study of conies, and that Euclid called them by the same name as that applied to corollaries because they were corollaries with respect to conies. But there appears to be no evidence to confirm this conjecture. S- The Surface4oci (tottoi -jrooi hirt^iaveia). The two books on this subject are mentioned by Pappus as part of the Treasury of Analysis 1 . As the other works in the list which were on plane subjects dealt only with straight lines, circles, and conic sections, it is a priori likely that among the loci in this treatise (loci which are surfaces) were included such loci as were cones, cylinders and spheres. Beyond this all is conjecture based on two lemmas given by Pappus in connexion with the treatise. (1) The first of these lemmas 1 and the figure attached to it are not satisfactory as they stand, but a possible restoration is indicated by Tannery*. If the latter is right, it suggests that one of the loci contained all the points on the elliptical parallel sections of a cylinder and was therefore an oblique circular cylinder. Other assumptions with regard to the conditions to which the lines in the figure may be subject would suggest that other loci dealt with were cones regarded as containing all points on particular elliptical parallel sections of the cones", (2) In the second lemma Pappus states and gives a complete proof of the focus-and-directrix property of a conic, viz. that the locus of a point whose distance front a given point is in a given ratio to its distance from a fixed line is a conic section, which is an ellipse, a parabola or a hyperbola according as the given ratio is less than, equal to, or greater than unity'. Two conjectures are possible as to the application of this theorem in Euclid's Surface-loci, {a) It may have been used to prove that the locus of a point whose distance from a given straight 1 Die Lekrcvwt den Kegilsiknitttn im Aliertttm, chapter VIII. * Pappus, vn. p. 636. " Hid- TO p- 1004. * Mullciin da seieneei math, eJ astro*., J* Sirie, VI. 1+9. 1 Further particulars will be found in The Works ef Archimedes, pp. litii — Ixiv, and in Zeuthen, Die Lthre von den KtgeUthttitun, p. 415 sqq. ' Pappus, Vtt. pp. j 006 — 1014, and Hultsdi's Appendix, pp. 1170 — 3. 1 6 INTRODUCTION [en. 11 line is in a given ratio to its distance from a given plane is a certain cone, (b) It may have been used to prove that the locus of a point whose distance from a given point is in a given ratio to its distance from a given plane is the surface formed by the revolution of a conic about its major or conjugate axis 1 . Thus Chasles may have been correct in his conjecture that the Surface-loci dealt with surfaces of revolution of the second degree and sections of the same". 6. The Conks. Pappus says of this lost work: "The four books of Euclid's Conies were completed by Apollonius, who added four more and gave us eight books of Conies'." It is probable that Euclid's work was lost even by Pappus' time, for he goes on to speak of "Aristaeus, who wrote the still extant five books of Solid Loci connected with the conies." Speaking of the relation of Euclid's work to that of Aristaeus on conies regarded as loci, Pappus says in a later passage (bracketed however by Huttsch) that Euclid, regarding Aristaeus as deserving credit for the discoveries he had already made in conies, did not (try to) anticipate him or construct anew the same system. We may no doubt conclude that the book by Aristaeus on solid loci preceded Euclid's on conies and was, at least in point of originality, more important Though both treatises dealt with the same subject-matter, the object and the point of view were different ; had they been the same, Euclid could scarcely have refrained, as Pappus says he did, from attempting to improve upon the earlier treatise. No doubt Euclid wrote on the general theory of conies as Apollonius did, but confined himself to those properties which were necessary for the analysis of the Solid Loci of Aristaeus. The Conks of Euclid were evidently superseded by the treatise of Apollonius. As regards the contents of Euclid's Conks, the most important source of our information is Archimedes, who frequently refers to propositions in conies as well known and not needing proof, adding in three cases that they are proved in the " elements of conies " or in "the conies," which expressions must clearly refer to the works of Aristaeus and Euclid 4 Euclid still used the old names for the conies (sections of a right- angled, acute 'angled, or obtuse-angled cone), but he was aware that an ellipse could be obtained by cutting a cone in any manner by a plane not parallel to the base (assuming the section to lie wholly between the apex of the cone and its base) and also by cutting a cylinder. This is expressly stated in a passage from the Pkaettontena of Euclid about to be mentioned". 7. The Pkaenometta. This is an astronomical work and is still extant. A much inter- 1 For further details see The Works of Arckimtd(s x pp. Ixiv, Lev, and Zeuthen, I. c. * Apcrfu hhtoriqtu, pp. 373 — 4. * Pappus, VII. p. 673. * For details of these propositions see my Apollonius of Perga, pp. xxxv, xxxvi. * Phattamtna, ed. Menge, p. 6: "if a cone or a cylinder be cut by a plane not parallel to the base, the section is a section of an acute-angled cone, which is like a shield \evftit)." ch. n] EUCLID'S OTHER WORKS 17 polated version appears in Gregory's Euclid. An earlier and better recension is however contained in the Ms. Vindobonensis philos. Gr. 103, though the end of the treatise, from the middle of prop. 16 to the last (18), is missing. The book, now edited by Menge 1 , consists of propositions in spheric geometry. Euclid based it on Autolycus' work Trepi Ktvov/ievTj? ecftatpas, but also, evidently, on an earlier text- book of Spkaerica of exclusively mathematical content. It has been conjectured that the latter textbook may have been due to Eudoxus'. 8. The Optics. This book needs no description, as it has been edited by Heiberg recently', both in its genuine form and in the recension by Theon. The Catoptrica published by Heiberg in the same volume is not genuine, and Heiberg suspects that in its present form it may be Theon's. It is not even certain that Euclid wrote Catoptrica. at all, as Proclus may easily have had Theon's work before him and inadvertently assigned it to Euclid 1 . 9. Besides the above-mentioned works, Euclid is said to have written the Elements of Music* (at tcara ftovatKrjv a"rotYet<a<r£ts). Two treatises are attributed to Euclid in our MSS. of the Mtfsici, the icararofii) Kavovos, Sectio canonis (the theory of the intervals), and the tta-aiyayTj appovi/ci} (introduction to harmony)*. The first, resting on the Pythagorean theory of music, is mathematical, and the style and diction as well as the form of the propositions mostly agree with what we find in the Elements. Jan thought it genuine, especially as almost the whole of the treatise (except the preface) is quoted in extenso, and Euclid is twice mentioned by name, in the commentary on Ptolemy's Harmonica published by Wallis and attributed by him to Porphyry. Tannery was of the opposite opinion'. The latest editor, Menge, sug- gests that it may be a redaction by a less competent hand from the genuine Euclidean Elements of Music. The second treatise is not Euclid's, but was written by Cleonides, a pupil of Aristoxenus*. Lastly, it is worth while to give the Arabians' list of Euclid's works. I take this from Suter's translation of the list of philosophers and mathematicians in the Fikrist, the oldest authority of the kind that we possess*. " To the writings of Euclid belong further [in addition to the Elements] : the book of Phaenomena ; the book of 1 Euelidis Optra omnia, vol. VIII., igib, pp. 2 — 156. 1 Heiberg, Euklid-Studien, p. 46 ; HtilEsch, Autolycut, p. xii ; A. A. Bjornbo, Studien iiber Menetaos' Spharik {Abhattdiungen nir Geschichte der wathematiscken tVissenscAaflen, Jtiv. 1002), p. ;6sqq. 3 Euelidis opera omnia, vol. VI [. (1895). * Heiberg, Euclid's Optics, tic. p. I. a Proclus, p. 69, 3. * Both trealises edited by Jan in Music i Scriptorts Gratci, 1895, pp. 113—166, 167 — 10;, and by Menge in Euelidis opera omnia, vol. VII]., 1916, pp. 157 — 183, 1S5 — 223. 7 Cpmptes rendus de VAcad. des inscriptions it belles-lettres, Paris, 1904, pp. +39 — 445. Cf. Bihtioiheea Mathtmaiica, VI3, [905-6, p. 225, note i. a Heiberg, Euktid-Studien, pp. 52 — 55 \ Jan, Musici Seriptores Graeci, pp. 169—174. 6 H. Suter, Das Mathematiker- Vcruichniss im Fikrist in Abhandlungen iter Gesckicktc der Mathematik, VI., 1892, pp. 1- — 87 (see especially p. 17). Cf. Casiri, I. 339, 340, and Gartz, Dc interpretiius et txplanatoribus Euelidis Arabieis, 1823, pp. 4, $• i8 INTRODUCTION [ch. n Given Magnitudes [Data] ; the book of Tones, known under the name of Music, not genuine ; the book of Division, emended by Thabit ; the book of Utilisations or Applications [Porisms], not genuine; the book of the Canon ; the book of the Heavy and Light ; the book of Synthesis, not genuine ; and the book of Analysis, not genuine." It is to be observed that the Arabs already regarded the book of Tones (by which must be meant the citrayayf} apuovticrj) as spurious. The book of Division is evidently the book on Divisions (of figures). The next book is described by Casiri as " liber de utilitate suppositus." Suter gives reason for believing the Porisms to be meant 1 , but does not apparently offer any explanation of why the work is supposed to be spurious. The book of the Canon is clearly the KaraTOfit} icavovos. The book on "the Heavy and Light" is apparently the tract Dt Uvi et ponderoso, included in the Basel Latin translation of 1537, and in Gregory's edition. The fragment, however, cannot safely be attributed to Euclid, for ( 1 ) we have nowhere any mention of his having written on mechanics, (2) it contains the notion of specific gravity in a form so clear that it could hardly be attributed to anyone earlier than Archimedes*. Suter thinks* that the works on Analysis and Synthesis (said to be spurious in the extract) may be further developments of the Data or Porisms, or may be the interpolated proofs of Euci. XIII. 1 — 5, divided into analysis and synthesis, as to which see the notes on those propositions. 1 Suter, op. tit. pp. 49, 50, Wen rich translated the word as " ulilia. " Suter says thai the nearest meaning of the Arabic word as of "porism" is tut, gain (Nutzen, Gewinn), while a further meaning is explanation, observation, addition : a gain arising out of what has preceded (cf. Prod us 1 definition of the po risen in the sense of a corollary). 1 Heiberg, Euklid-Stvditn, pp. 0, 10. * Suter, ?p* til. p. 50. CHAPTER III. GREEK COMMENTATORS ON THE ELEMENTS OTHER THAN PROCLUS. That there was no lack of commentaries on the Elements before the time of Proctus is evident from the terms in which Proclus refers to them ; and he leaves as in equally little doubt as to the value which, in his opinion, the generality of them possessed. Thus he says in one place (at the end of his second prologue) 1 : " Before making a beginning with the investigation of details, I warn those who may read me not to expect from me the things which have been dinned into our ears ad nauseam (St,aT€9pv\*)T<u) by those who have preceded me, viz. lemmas, cases, and so forth. For I am surfeited with these things and shall give little attention to them. But I shall direct my remarks principally to the points which require deeper study and contribute to the sum of philosophy, therein emulating the Pythagoreans who even had this common phrase for what I mean ' a figure and a platform, but not a figure and sixpence 1 .' " In another place' he says : " Let us now turn to the elucidation of the things proved by the writer of the Elements, selecting the more subtle of the comments made on them by the ancient writers, while cutting down their interminable diffuseness, giving the things which are more systematic and follow scientific methods, attaching more importance to the working-out of the real subject-matter tharr to the variety of cases and lemmas to which we see recent writers devoting themselves for the most part." At the end of his commentary on Eucl, I. Prockis remarks' that the commentaries then in vogue were full of all sorts of confusion, and contained no account of causes, no dialectical discrimination, and no philosophic thought. These passages and two others in which Proclus refers to "the commentators*" suggest that these commentators were numerous. He does not however give many names; and no doubt the only important commentaries were those of Heron, Porphyry, and Pappus. 1 Proclus, p. 84, 8. ■ i.e. we reach a certain height, use the platform so attained as a base on which to build another stage, then use that as a base and so on. ' Proclus, p. 300, to. 4 Hid. p. 431, 15. * ibid. p. 389, \\ \ p* 318, r6. io INTRODUCTION [ch. hi I. Heron. Froctus alludes to Heron twice as Heron mecAanicus 1 , in another place* he associates him with Ctesibius, and in the three other passages ' where Heron is mentioned there is no reason to doubt that the same person is meant, namely Heron of Alexandria. The date of Heron is still a vexed question. In the early stages of the controversy much was made of the supposed relation of Heron to Ctesibius. The Dest MS. of Heron's Belopoeica has the heading "Hpowo? Kryvi&iov 0e\oTrotiicd, and an anonymous Byzantine writer of the tenth century, evidently basing himself on this title, speaks of Ctesibius as Heron's xafljfyjjTTJf, " master " or " teacher." We know of two men of the name of Ctesibius. One was a barber who lived in the time of Ptolemy Euergetes II, i.e. Ptolemy VII, called Physcon {died 117 B.C.), and who is said to have made an improved water-organ*. The other was a mechanician mentioned by Athenaeus as having made an elegant drinking-horn in the time of Ptolemy Philadelphus (285-247 B.C.)*. Martin* took the Ctesibius in question to be the former and accord- ingly placed Heron at the beginning of the first century B.C., say 1 26-50 B.C But Philo of Byzantium 7 , who repeatedly mentions Ctesi- bius by name, says that the first mechanicians had the advantage of being under kings who loved fame and supported the arts. Hence our Ctesibius is more likely to have been the earlier Ctesibius who was contemporary with Ptolemy 1 1 Philadelphus. But, whatever be the date of Ctesibius, we cannot safely conclude that Heron was his immediate pifpil. The title " Heron's (edition of) Ctesibius's Belopoeica" does not, in fact, justify any inferenee as to the interval of time between the two works. We now have better evidence for a terminus post quern. The Metrica of Heron, besides quoting Archimedes and Apollonius, twice refers to " the books about straight lines (chords) in a circle " {iv rots wept r£* 4» xvtcXqt eu&ei&p). Now we know of no work giving a Table of Chords earlier than tbat of Hipparchus. We get, therefore, at once, 1 50 B.c. or thereabouts as the terminus post quern. But, again, Heron's Meckanica quotes a definition of " centre of gravity " as given by " Posidonius, a Stoic " : and, even if this Posidonius lived before Archimedes, as the context seems to imply, it is certain that another work of Heron's, the Definitions, owes something to Posidonius of Apamea or Rhodes, Cicero's teacher (135—51 B.C.). This brings Heron's date down to the end of the first century B.C., at least We have next to consider the relation, if any, between Heron and Vitruvius. In his De Architecture brought out apparently in 14 B.C., Vitruvius quotes twelve authorities on machinationes including Archytas 1 Proclus, p. 305, 14 ; p. 346, 13. 1 ibid. p. 41, to. * ibid. p. 196, 16 ; p. 313, 7 : p- 4*9, 13. * Athetuetu, Ddpno-Sefh. iv., c. 7;, p. 174 *— <■ • * ibid, xi., c. 97, p. 497 b — e. * Martin, KschtriJui mr lavitetUt mtvritgta d'Hiren d ' Altxandrit, Puii, l8j+, p. 17. 7 Philo, Mi/than. Synt., p. 50, 38, ed. Schone. ch. in] GREEK COMMENTATORS OTHER THAN PROCLUS ai (second), Archimedes (third), Ctesibius (fourth) and Philo of Byzan- tium (sixth), but does not mention Heron. Nor is it possible to establish inter-dependence between Vitmvius and Heron ; the differ- ences between them seem on the whole more numerous and important than the resemblances (e.g. Vitruvius uses 3 as the value of tt, while Heron always uses the Archimedean value 3^). The inference is that Heron can hardly have written earlier than the first century A.D. The most recent theory of Heron's date makes him later than Claudius Ptolemy the astronomer (i 00- 1 78 A.D.). The arguments are mainly these. (!) Ptolemy claims as a discovery of his own a method of measuring the distance between two places (as an arc of a great circle on the earth's surface) in the case where the places are neither on the same meridian nor on the same parallel circle. Heron, in his Dtoptra, speaks of this method as of a thing generally known to experts. (2) The dioptra described in Heron's work is a fine and accurate instrument, much better than anything Ptolemy had at his disposal. (3) Ptolemy, in his work Ilepi poir&v, asserted that water with water round it has no weight and that the diver, however deep he dives, does not feel the weight of the water above him. Heron, strangely enough, accepts as true what Ptolemy says of the diver, but is dissatisfied with the explanation given by "some," namely that it is because water is uniformly heavy — this seems to be equivalent to Ptolemy's dictum that water in water has no weight — and he essays a different explanation based on Archimedes. (4) It is suggested that the Dionysius to whom Heron dedicated his Definitions is a certain Dionysius who was firaefectus itrbi in 301 A.D. On the other hand Heron was earlier than Pappus, who was writing under Diocletian (284-305 A.D.), for Pappus alludes to and draws upon the works of Heron. The net result, then, of the most recent research is to place Heron in the third century a.d. and perhaps little earlier than Pappus. Heiberg 1 accepts this conclusion, which may therefore, perhaps, be said to hold the field for the present*. That Heron wrote a systematic commentary on the Elements might be inferred from Proclus, but it is rendered quite certain by references to the commentary in Arabian writers, and particularly in an-Nairlzi's commentary on the first ten Books of the Elements. The Fihrist says, under Euclid, that " Heron wrote a commentary on this book [the Elements], endeavouring to solve its. difficulties' " ; and under Heron, " He wrote : the book of explanation of the obscurities in Euclid'...." An-Nairfzl's commentary quotes Heron by name very frequently, and often in such a way as to leave no doubt that the author had Heron's work actually before him. Thus the extracts are given in the first person, introduced by "Heron says" ("Dixit Yrinus" 1 Htronis Alexandrini o/vra t vol. V. (Teuhner, 1*914), p. ix. - Fuller details of the various arguments will be found in my History of Greek Maiht- matin, toil, vol. it., pp. j 96 —306. 3 Das Matkematiker.VeneitAniss im Fihrist (tr. Sutcr), p. 16. * Hid. p. 11, t a INTRODUCTION [ch. in or " Heron ") ; and in other places we are told that Heron " says nothing," or * is not found to have said anything," on such and such a proposition. The commentary of an-Nairlzi is in part edited by Besthorn and Heiberg from a Leiden MS. of the translation of the Elements by al-Hajjaj with the commentary attached 1 . But this MS. only contains six Books, and several pages in the first Book, which contain the comments of Simplicius on the first twenty-two defini- tions of the first Book, are missing. Fortunately the commentary of an-Nairlzi has been discovered in a more complete form, in a Latin translation by Gherardus Cremonensis of the twelfth century, which contains the missing comments by Simplicius and an-Nairlzl's com- ments on the first ten Books. This valuable work has recently been edited by Curtze'. Thus from the three sources, P rod us, and the two versions of an-Nairlzi, which supplement one another, we are able to form a very good idea of the character of Heron's commentary. In some cases observations given by Proclus without the name of their author are seen from an-Nairizi to be Heron's ; in a few cases notes attributed by Procius to Heron are found in an-Nairizi without Heron's name; and, curiously enough, one alternative proof (of I. 25) given as Heron's by Proclus is introduced by the Arab with the remark that he has not been able to discover who is the author. Speaking generally, the comments of Heron do not seem to have contained much that can be called important. We find (1) A few general notes, e.g. that Heron would not admit more than three axioms. (2) Distinctions of a number of particular cases of Euclid's pro- positions according as the figure is drawn in one way or in another. Of this class are the different cases of 1. 35, 36, HI. 7, 8 (where the chords to be compared are drawn on different sides of the diameter instead of on the same side), III. 12 (which is not Euclid's, but Heron's own, adding the case of external contact to that of internal contact in III. 1 1), VI. 19 (where the triangle in p which an additional line is drawn is taken to be the smaller of the two), Vii. 19 (where he gives the particular case of three numbers in continued proportion, instead of four proportionals). (3) Alternative proofs. Of these there should be mentioned (a) the proofs of H. I — 10 "without a figure," being simply the algebraic forms of proof, easy but un instructive, which are so popular nowadays, the proof of III. 25 (placed after III. 30 and starting from the arc instead of the chord), III. 10 (proved by 111. 9), 111. 13 (a proof preceded by a lemma to the effect that a straight line cannot meet a circle in more than two points). Another class of alternative proof is 1 Cedes Leidtnris 399, r. Eudidis Eiementa ex interprets itme ai-ffadschdiehadsekii cum commentariis ai-Naritii. Five parts carrying I he work Lo the end of Book IV. were issued in 1893. 1897, J900, 1905 awl 1910 respectively. * Anaritii in decern Hbros prieres elementorum Eudidis eonimtntarii ex ittterpreiaiione Gherardi Cremonensis... edidit Maximilian us Curtze (Teubner, Leipzig, 1899}. ch. in] GREEK COMMENTATORS OTHER THAN PROCLUS 33 (6) that which is intended to meet a particular objection («Wra«-«) which had been or might be raised to Euclid's construction. Thus in certain cases he avoids producing a particular straight line, where Euclid produces it, in order to meet the objection of any one who should deny our right to assume that there is any space available^. Of this class are Heron's proofs of I. 1 1, 1, 20, and his note on I. 16. Similarly on I. 48 he supposes the right-angled triangle which is constructed to be constructed on the same side of the common side as the given triangle is. A third class (c) is that which avoids redttctio ad absurdum. Thus, instead of indirect proofs, Heron gives direct proofs of I. 19 (for which he requires, and gives, a preliminary lemma), and of I. 25. (4) Heron supplies certain converses of Euclid's propositions, e.g. converses of II. 12, 13, vm. 27. (5) A few additions to, and extensions of, Euclid's propositions are also found. Some are unimportant, e.g. the construction of isosce'es and scalene triangles in a note on I. 1 , the construction of tivo tangents in in. 17, the remark that vn. 3 about finding the greatest common measure of three numbers can be applied to as many numbers as we please (as Euclid tacitly assumes in VII. 31). The most important extension is that of in. 20 to the case where the angle at the circumference is greater than a right angle, and the direct deduction from this extension of the result of III. 22. Interesting also are the notes on I. 37 (on I. 24 in Proclus), where Heron proves that two triangles with two sides of one equal to two sides of the other and with the included angles supplementary are equal, and compares the areas where the sum of the two included angles (one being supposed greater than the other) is less or greater than two right angles, and on I, 47, where there is a proof (depending on preliminary lemmas) of the fact that, in the figure of the proposition, the straight lines AL, BK, CF meet in a point. After IV. 16 there is a proof that, in a regular polygon with an even number of sides, the bisector of one angle also bisects its opposite, and an enunciation of the corresponding proposition for a regular polygon with an odd number of sides. Van Peach" gives reason for attributing to Heron certain other notes found in Proclus, viz. that they are designed to meet the same sort of points as Heron had in view in other notes undoubtedly written by him. These are (a) alternative proofs of I. 5, I. 17, and I. 32, which avoid the producing of certain straight lines, (b) an alternative proof of I. 9 avoiding the construction of the equilateral triangle on the side of BC opposite to A ; {c) partial converses of 1. 35 — 38, starting from the equality of the areas and the fact of the parallelograms or triangles being in the same parallels, and proving that the bases are the same or equal, may also be Heron's. Van Pesch further supposes that it was in Heron's commentary that the proof by Menelaus of I, 25 and the proof by Philo of I. 8 were given. 1 Cf. Proclui, it;., 1 tlU W701 tii rlrxov pj) eliivni..., 189, 18 \tyct oir Tti Sri ott tm riwn,,., 3 Dt Prpcli jontibusi Lugduni-fiatavomm, 1900, a 4 INTRODUCTION [ch. hi The last reference to Heron made by an-NairtzI occurs in the note on VIII. 27, so that the commentary of the former must at least have reached that point. II. Porphyry. The Porphyry here mentioned is of course the Neo-Flatonist who lived about 232-304 a.d. Whether he really wrote a systematic commentary on the Elements is uncertain. The passages in Proclus which seem to make this probable are two in which he mentions him ( 1 ) as having demonstrated the necessity of the words " not on the same side " in the enunciation of I. 14 1 , and {2) as having pointed out the necessity of understanding correctly the enunciation of I, 26, since, if the particular injunctions as to the sides of the triangles to be taken as equal are not regarded, the student may easily fall into error*. These passages, showing that Porphyry carefully analysed Euclid's enunciations in these cases, certainly suggest that his remarks were part of a systematic commentary. Further, the list of mathematicians in the Fihrist gives Porphyry as having written "a book on the Elements." It is true that Wen rich takes this book to have been a work by Porphyry mentioned by Suidas and Proclus ( Theolog. Platan.), •jrtpl dpx&v libri II.' There is nothing of importance in the notes attributed to Porphyry by Proclus. ( 1) Three alternative proofs of I. 20, which avoid producing a side of the triangle, are assigned to Heron and Porphyry without saying which belonged to which. If the first of the three was Heron's, I agree with van Pesch that it is more probable that the two others were both Porphyry's than that the second was Heron's and only the third Porphyry's. For they are similar in character, and the third uses a result obtained in the second*. (2) Porphyry gave an alternative proof of I. 18 to meet a childish objection which is supposed to require the part of A C equal to A B to be cut off from CA and not from A C. Proclus gives a precisely similar alternative proof of I. 6 to meet a similar supposed objection ; and it may well be that, though Proclus mentions no name, this proof was also Porphyry's, as van Pesch suggests*. Two other references to Porphyry found in Proclus cannot have anything to do with commentaries on the Elements. In the first a work called the Xv^fuxrd is quoted, while in the second a philo- sophical question is raised. III. Pappus. The references to Pappus in Proclus are not numerous; but we have other evidence that he wrote a commentary on the Elements. Thus a scholiast on the definitions of the Data uses the phrase " as 1 Proclus, pp. 197, 1—198, 10. ' ibid, p. 351, 13, 141ml the pages preceding. * Fihrist (tr. Suter), p. 9, jo and p. 45 {note j). * Van Pesch, Di Pretli fcntibus, pp. no, 130. Heibetg assigned them as above in hit Eu&lid-Studitn (p. 160), but seems to have changed hit view later. (See Besthom- Heiberg, Codex Ltidatsis, p. 93, note ».) * Van Pesch, sp. cii. pp. 130 — \. ch. tit] GREEK COMMENTATORS OTHER THAN PROCLUS as Pappus says at the beginning of his (commentary) on the loth (book) of Euclid 1 ." Again in the Fihrist we are told that Pappus wrote a commentary to the tenth book of Euclid in two parts'. Fragments of this still survive in a MS. described by Woepcke*, Paris. No. 952. 2 (supplement arabe de la Bibliotheque imperiale), which contains a translation by Abu 'Uthman (beginning of [Oth century) of a Greek commentary on Book X. It is in two books, and there can now be no doubt that the author of the Greek commentary was Pappus 4 . Again Eutocius, in his note on Archimedes, On the Sphere and Cylinder I. r3, says that Pappus explained in his commentary on the Elements how to inscribe in a circle a polygon similar to a polygon inscribed in another circle ; and this would presumably come in his commentary on Book XII., just as the problem is solved in the second scholium on Eucl, XII. 1. Thus Pappus' commentary on the Elements must have been pretty complete, an additional confirmation of this supposition being forthcoming in the reference of Marin us (a pupil and follower of Proclus) in his preface to the Data to " the com- mentaries of Pappus on the book*." The actual references to Pappus in Proclus are as follows : (1) On the Postulate (4) that all right angles are equal, Pappus is quoted as saying that the converse, viz. that all angles equal to a right angle are right, is not true', since the angle included between the arcs of two semicircles which are equal, and have their diameters at right angles and terminating at one point, is equal to a right angle, but is not a right angle. (2) On the axioms Pappus is quoted as saying that, in addition to Euclid's axioms, others are on record as well (ovvavarfpafao-Sai) about unequals added to equals and equals added to unequals 1 ; these, says Proclus, follow from the Euclidean axioms, while others given by Pappus are involved by the definitions, namely those which assert that "all parts of the plane and of the straight line coincide with one another," that " a point divides a straight line, a line a surface, and a surface a solid," and that "the infinite is (obtained) in magnitudes both by addition and diminution*." 1 Euclid's Data, ed. Menge. p. 761. ' fihrist (tr. Sitter), p. It, * JlfJmeires prlsenth A Faeadimit ties jcieneej, 1856, Kiv. pp. 658 — 7(0. * Woepcke read the name of the author, in the Litle of the first book, as B.tes (the dot representing 11 missing vowel). He quotes also from other MSS. (e.g. of the Ta*rikh al- HttlamA and of the Fihrist) when he reads the name of the commentator as B .lis, B .n.t at B.I.j, Woepcke takes this author to be Valens, and thinks it possible that he may be the same as the astrologer Vettius Valens. This Heiberg (Eu&tiii-Studun, pp. 169, 170) proves to be impossible, because, while one of the mss. quoted by Woepcke says that "B.n.j, le Rs&mV (late- Greek) was later than Claudius Ptolemy and the Fihrist says 11 B.I. j, le Betsmt" wrote a commentary on Ptolemy's Plastisphaerium, Vettius Valens seems to have Lived under Hadrian, and must therefore have been an elder contemporary of Ptolemy. But Suter shows (Fihrist, p. at and p. j*, note 91) that Bants is only distin- guished from Babe j by the position of a certain dot, and Bates may also easily have arisen from an original Boies (there is no P in Arabic), so that Pappus must be the person meant. This is further confirmed by the fact that the Fihrist gives this author and Valens as the subjects of twu separate paragraphs, attributing to the latter astrological works only. * Heiberg, Euklid-Studien, p. 173; Euclid's Data, ed. Menge, pp. 356, lii. * Proclus, pp. 189, jqo. 7 ibid. p. 197, 6—10. * Hid. p. 198, 3— ij. 16 INTRODUCTION [ch, hi (3) Pappus gave a pretty proof of I. 5. This proof has, I think, been wrongly understood ; on this point see my note on the proposition. (4) On I. 47 Proclus says 1 : " As the proof of the writer of the Elements is manifest, I think that it is not necessary to add anything further, but that what has been said is sufficient, since indeed those who have added more, like Heron and Pappus, were obliged to make use of what is proved in the sixth book, without attaining any important result." We shall see what Heron's addition consisted of; what Pappus may have added we do not know, unless it was some- thing on the lines of his extension of I. 47 found in the Synagoge (iv. p. 176, ed. Hultsch). We may fairly conclude, with van Pesch", that Pappus is drawn upon in various other passages of Proclus where he quotes no authority, but where the subject-matter reminds us of other notes expressly assigned to Pappus or of what we otherwise know to have been favourite questions with him. Thus : [. We are reminded of the curvilineal angle which is equal to but not a right angle by the note on I. 32 to the effect that the converse (that a figure with its interior angles together equal to two right angles is a triangle) is not true unless we confine ourselves to rectilineal figures. This statement is supported by reference to a figure formed by four semicircles whose diameters form a square, and one of which is turned inwards while the others are turned outwards. The figure forms two angles "equal to" right angles in the sense described by Pappus on Post. 4, while the other curvilineal angles are not considered to be angles at all, and are left out in summing the internal angles. Similarly the allusions in the notes on I. 4, 23 to curvilineal angles of which certain moon -shaped angles {i^vouZwi) are shown to be "equal to" rectilineal angles savour of Pappus. 2. On 1. 9 Proclus says* that " Others, starting from the Archi- medean spirals, divided any given rectilineal angle in any given ratio." We cannot but compare this with Pappus IV. p. 286, where the spiral is so used ; hence this note, including remarks immediately preceding about the conchoid and the quad rat rix, which were used for the same purpose, may very well be due to Pappus. 3, The subject of isoperi metric figures was a favourite one with Pappus, who wrote a recension of Zenodorus' treatise on the subject 1 . Now on I. 35 Proclus speaks 1 about the paradox of parallelograms having equal area (between the same parallels) though the two sides between the parallels may be of any length, adding that of parallelo- grams with equal perimeter the rectangle is greatest if the base be given, and the square greatest if the base be not given etc. He returns to the subject on I. 37 about triangles'. Compare 7 also his note on I. 4. These notes may have been taken from Pappus. 1 Proclus, p. 439, — 15. 9 Vin Pesch, Dt Prodi fsntitws, p. 13+ sqq. ' Proclus, p. 371, 10. * Pappus, v. pp. 304 — 330 ; for Zenodorus own treatise see Hultsch's Appendix, pp. 1 189 — 1*11. * Proclus, pp. 396—8. ■ ibvl, pp. 403— 4. T ibid. pp. 136—7. ch. in] GREEK COMMENTATORS OTHER THAN PROCLUS 27 4. Again, on I. 21, Procius remarks on the paradox that straight lines may be drawn from the base to a point within a triangle which are (1) together greater than the two sides, and (2) include a less angle, provided that the straight lines may be drawn from points in the base other than its extremities. The subject of straight lines satisfying condition (1) was treated at length, with reference to a variety of cases, by Pappus 1 , after a collection of "paradoxes" by Erycinus, of whom nothing more is known. Procius gives Pappus' first case, and adds a rather useless proof of the possibility of drawing straight lines satisfying condition {2) alone, adding that " the proposi- tion stated has been proved by me without using the parallels of the commentators'." By "the commentators" Pappus is doubtless meant 5. Lastly, the "four-sided triangle," called by Zenodorus the " hollow-angled,"* is mentioned in the notes on I. Def. 24 — 29 and t. 21. As Pappus wrote on Zenodorus' work in which the term occurred*, Pappus may be responsible for these notes. IV. Simplicius. According to the FihrisP, Simplicius the Greek wrote "a com- mentary to the beginning of Euclid's book, which forms an introduc- tion to geometry." And in fact this commentary on the definitions, postulates and axioms (including the postulate known as the Parallel- Axiom) is preserved in the Arabic commentary of an-Nairizi*. On two subjects this commentary of Simplicius quotes a certain "Aganis," the first subject being the definition of an angle, and the second the definition of parallels and the parallel-postulate. Simplicius gives word for word, in a long passage placed by an-Nairizi after I. 29, an attempt by " Aganis " to prove the para lie I -postulate. It starts from a definition of parallels which agrees with Geminus' view of them as given by Procius', and is closely connected with the definition given by Posidonius 8 . Hence it has been assumed that " Aganis " is none other than Geminus, and the historical importance of the commentary of Simplicius has been judged accordingly. But it has been recently shown by Tannery that the identification of "Aganis" with Geminus is practically impossible 11 . In the translation of Besthorn-Heiberg Aganis is called by Simplicius in one place "philosophus Aganis," in another " magister noster Aganis," in Gherard's version he is " socius Aganis" and "socius noster Aganis." These expressions seem to leave no doubt that Aganis was a contemporary and friend, if not master, of Simplicius ; and it is impossible to suppose that Simplicius (fl. about 500 a.d.) could have used them of a man who lived four and 1 Pappus, HI. pp. 104 — 1 jo. 3 Procius, p. 318, 15. 1 Procius, p. |6{, »+; ef. pp. 318, 319. * See Pappus, ed. Hultsch, pp. 1154, iip6. 1 Fihrist (tr. Suter), p. 11. 1 An-NairM, ed. Bestbom-Heiberg, pp. 9 — 41, 119— 133, ed. Curtie, pp. 1 — 37,65 — 73. The Cedtx Ltittensis, from which Besthorn and Heiberg's edition is taken, has unfortunately lost some leaves, so that there is a gap from Def. 1 to Def. 13 (parallels). The loss is, how- ever, made good by Curtse's edition of the translation by Gherard of Cremona. 7 Procius, p. 177, 11. 8 ibid, p. 176, 7. * Bibliothtca Afathtmatka, ITj, 1900, pp. o — 11. 28 INTRODUCTION [ch. hi a half centuries before his time. A phrase in Simplicius' word-for- word quotation from Aganis leads to the same conclusion. He speaks of people who objected " even in ancient times " (iam antiquitus) to the use by geometers of this postulate. This would not have been an appropriate phrase had Gem in us been the writer. I do not think that this difficulty can he got over by Sulcus suggestion 1 that the passages in question may have been taken out of Heron's commentary, and that an-Nairlzl may have forgotten to name the author ; it seems clear that Simplicius is the person who described " Aganis." Hence we are driven to suppose that Aganis was not Geminus, but some unknown contemporary of Simplicius 5 . Considerable interest will however continue to attach to the comments of Simplicius so fortunately preserved. Proclus tells us that one Aegaeas {? Aenaeas) of Hi era polls wrote an epitome of the Elements* ; but we know nothing more of him or of it 1 Ztitsckrififiir Math. u. Pkysik, XLIV., hiat.-litt, Ablh. p. fit. * The above argument seems to me quite insuperable. The other arguments of Tannery do not, however, cany conviction to my mind. I do not follow the reasoning based on Aganis' definition of an angle. It appears to me a pure assumption that Geminus would have seen that Posidonius' definition of parallels was not admissible. Nor does it seem to me to count for much that Proclus, while telling us that Geminus held that the postulate ought to be proved and warned the unwary against hastily concluding that two straight lines approaching one another must necessarily meet (cf, a curve and its asymptote), gives no hint that Geminus did try to prove the postulate. It may well be that Proclus omitted Geminus' " proof" (if he wrote one) because he preferred Ptolemy's attempt which he give* (pp. 36S — 7r- * Proelui, p. 361, *i. CHAPTER IV. PROCLUS AND HIS SOURCES'. It is well known that the commentary of P roc 1 us on Eucl. Book i. is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus. They are the more precious because the original works of the forerunners of Euclid, Archimedes and Apollonius are lost, having probably been discarded and forgotten almost immediately after the appearance of the masterpieces of that great trio. Prod us himself lived 410-485 AD, so that there had already passed a sufficient amount of time for the tradition relating to the pre -Euclidean geometers to become obscure and defective. In this connexion a passage is quoted from Simplkius' who, in his account of the quadrature of certain lunes by Hippocrates of Chios, while mentioning two authorities for his statements, Alexander Aphro- disiensis (about 220 A.D.) and Eudemus, says in one place *, " As regards Hippocrates of Chios we must pay more attention to Eudemus, since he was nearer the times, being a pupil of Aristotle:" The importance therefore of a critical examination of Proclus' commentary with a view to determining from what original sources he drew need not be further emphasised. Proclus received his early training in Alexandria, where Olympio- dorus was his instructor in the works of Aristotle, and mathematics was taught him by one Heron' (of course a different Heron from the •' mechanicus Hero" of whom we have already spoken). He after- wards went to Athens where he was imbued by Plutarch, and by Syrianus, with the Neo- Platonic philosophy, to which he then devoted 1 My task in this chapter is nude easy by the appearance, in the nick of time, of the dissertation Dt Froili fontibus by J. G, van Pescb (Lugduni-Batavoruro, A pud L. van Nifterik, MDCCCC). The chapters dealing directly with the subject show a thorough acquaintance on the part of the author with alt the literature hearing on it; he coven the whole field and he exercises a sound and sober judgment in forming his conclusion s. The same cannot always be said of his only predecessor in the same inquiry, Tannery (in La Ghmitrit grtcqut, 1887), who often robs his speculations of much of their value through his proneness to run away with on idea; be docs so in ihi* case, basing most of his conclusions on an arbitrary and unwarranted assumption as to the significance of the words ot wtpt rtira (e.g. 'Hpvra, UoettSwrior etc) as used in Proclus. ■ Simplicius on Aristotle's Phyiits, ed. Diels, pp. j+ — 69. ' Hid. p. 68, 31. 4 Cf. Martin, Rtsktrchts atr la vu if la nonages d'Hinm iTAltxandrit, pp. 140 — 1. * 3 o INTRODUCTION [ch. iv heart and soul, becoming one of its most prominent exponents. He speaks everywhere with the highest respect of his masters, and was in turn regarded with extravagant veneration by his contem- poraries, as we learn from Marinus his pupil and biographer. On the death of Syrianus he was put at the head of the Neo-Platonic school. He was a man of untiring industry, as is shown by the number of books' which he wrote, including a large number of com- mentaries, mostly on the dialogues of Plato. He was an acute dialectician, and pre-eminent among his contemporaries in the range of his learning 1 ; he was a competent mathematician ; he was even a poet. At the same time he was a believer in all sorts of myths and mysteries and a devout worshipper of divinities both Greek and Oriental. Though he was a competent mathematician, he was evidently much more a philosopher than a mathematician 1 . This is shown even in his commentary on Eucl. L, where, not only in the Prologues (especially the first), but also in the notes themselves, he seizes any opportunity for a philosophical digression. He says himself that he attaches most importance to "the things which require deeper study and contribute to the sum of philosophy 3 "; alternative proofs, cases, and the like (though he gives many) have no attraction for him ; and, in particular, he attaches no value to the addition of Heron to I, 47*, which is of considerable mathematical interest Though he esteemed mathematics highly, it was only as a handmaid to philosophy He quotes Plato's opinion to the effect that "mathematics, as making use of hypotheses, falls short of the non- hypothetical and perfect science 6 ". .."Let us then not say that Plato excludes mathematics from the sciences, but that he declares it to be secondary to the one supreme science'." And again, while "mathematical science must be considered desirable in itself, though not with reference to the needs of daily life," "if it is necessary to refer the benefit arising from it to something else, we must connect that benefit with intellectual know- ledge (voepitv yvaxrtv), to which it leads the way and is a propaedeutic, clearing the eye of the soul and taking away the impediments which the senses place in the way of the knowledge of universals (r&p $\an>y." We know that in the Neo-Platonic school the younger pupils learnt mathematics ; and it is clear that Proclus taught this subject, and that this was the origin of the commentary. Many passages show him as a master speaking to scholars. Thus "we Have illustrated 1 Zeller calls him "Der Gelehrtc, dem kern Feld damaligen Wissens verschlossen ist." * Van Pesch observes that in his commentaries on the Timaeus (pp. 671 — 2) he speaks as no real mathematician could have spoken. In the passage referred to the question is whether the sun occupies a middle place among the planets. Proclus rejects the view of Hipparchus and Ptolemy because "& Vtwpy6t" (sc. the Chaldean, says Zeller} thinks otherwise, "whom it is not lawful to disbelieve-" Martin says rather neatly, " Pour Proclus, leg Elements d'Euclide ont l 1 hen reuse chance de n'elre contredits ni par fes Oracles chaldalques, ibi par les speculations des pythagoriciens anciens et nouveaux ,.,,,." * Proclus, p. 84, 13. ' ibid. p. 429, tit, 6 ibid. p. 31, 20. ■ ibid. p. 31, «. 7 ibid. p. 37, J7 to 18, 7; cf. also p, »i, 15, pp. 46, +7. ch. iv] PROCLUS AND HIS SOURCES 31 and made plain all these things in the case of the first problem, but it is necessary that my hearers should make the same inquiry as regards the others as well '," and " I do not indicate these things as a merely incidental matter but as preparing us beforehand for the doctrine of the Timaeus 1 .* 1 Further, the pupils whom he was addressing were beginners in mathematics ; for in one place he says that he omits "for the present" to speak of the discoveries of those who employed the curves of Nicomedes and Hippias for trisecting an angle, and of those who used the Archimedean spiral for dividing an angle in any given ratio, because these things would be too difficult for beginners (Svff&ewpijTovs to« tla-ayoftivoi^)'. Again, if his pupils had not been beginners, it would not have been necessary for Proclus to explain what is meant by saying that sides subtend certain angles', the difference between adjacent and vertical angles' etc., or to exhort them, as he often does, to work out other particular cases for themselves, for practice {yvpvaaias ihteica)'. The commentary seems then to have been founded on Proclus' lectures to beginners in mathematics. But there are signs that it was revised and rt -edited for a larger public ; thus he gives notice in one place' "to those who shall come upon" his work (roh ivrev^o- fiivov;). There are also passages which could not have heen under- stood by the beginners to whom he lectured, e.g. passages about the cylindrical helix', conchoids and cissoids*. These passages may have been added in the revised edition, or, as van Pesch conjectures, the explanations given in the lectures may have been much fuller and more comprehensible to beginners, and they may haw; been shortened on revision. In his comments on the propositions of Euclid, Proclus generally proceeds in this way : first he gives explanations regarding Euclid's proofs, secondly he gives a few different cases, mainly for the sake of practice, and thirdly he addresses himself to refuting objections raised by cavillers to particular propositions. The latter class of note he deems necessary because of "sophistical cavils" and the attitude of the people who rejoiced in finding paralogisms and in causing annoyance to scientific men". His commentary does not seem to have been written for the purpose of correcting or improving Euclid. For there are very few passages of mathematical content in which Proclus can be supposed to be propounding anything of his own ; nearly all are taken from the works of others, mostly earlier commentators, so that, for the purpose of improving on or correcting Euclid, there was no need for his commentary at all. Indeed only in one place does he definitely bring forward anything of his own to get over a difficulty which he finds in Euclid"; this i, c where he tries to I Proclm, p. no, 1 9. ' il/id. p. 384, a. * ibid. p. 171, 11. * ibid, p. 338, 1*. * ibid.?. 19S, 14. * Cf. p. 114, is {on i. »). * ibid. p. 84, 9. ' ibid- p. 105. * ibid. p. 1 ti. " 'bid. p. 375, 9. II ibid. pp. 368—373. 32 INTRODUCTION [ch. iv prove the parallel-postulate, after first giving Ptolemy's attempt and then pointing out objections to it. On the other hand, there are a number of passages in which he extols Euclid; thrice 1 also he supports Euclid against Apollonius where the latter had given proofs which he considered better than Euclid's (I. 10, II, and 23). Allusion must be made to the debated question whether Proclus continued his commentaries beyond Book I, His intention to do so is clear from the following passages. Just after the words above quoted about the trisection etc. of an angle by means of certain curves he says, " For we may perhaps more appropriately examine these things on the third book, where the writer of the Elements bisects a given circumference*." Again, after saying that of all parallelograms which have the same periifieter the square is the greatest " and the rhomboid least of all," he adds : "But this we will prove in another place ; for it is more appropriate to the (discussion of the) hypotheses of the second book'." Lastly, when alluding (on I. 45) to the squaring of the circle, and to Archimedes' proposition that any circle is equal to the right-angled triangle in which the perpendicular is equal to the radius of the circle and the base to its perimeter, he adds, "But of this elsewhere* " ; this may imply, an intention to treat of the subject on Eucl. XII., though Heiberg doubts It*. But it is clear that, at the time when the commentary on Book I. was written, Proclus had not yet begun to write on the other Books and was uncertain whether he would be able to do so : for at the end he says", " For my part, if I should be able to discuss the other books' in the same manner, I should give thanks to the gods ; but, if other cares should draw me away, I beg those who are attracted by this subject to complete the exposition of the other books as well, following the same method, and addressing themselves throughout to the deeper and better defined questions involved " (to irpayfunetaiBe<s iravraxov /cat evhiaLperov nerahttoKOvTfx;). There is in fact no satisfactory evidence that Proclus did actually write any more commentaries than that on Book I." The contrary view receives support from two facts pointed out by Heiberg, viz. (1) that the scholiast's copy of Proclus was not so much better than our 1 Proclus, p. 1S0, 91 p. 183, 10; pp. 335, 3j6. 2 ibid, p. 373, 14. ' ibid. p. 39.8, i8. • ibid. p. 41J, 6. * Heiberg, EailidStuditn, p. [65, note. * Proclus, p. 431, g, T The words in the Greek are : */ p£p ivvTjSditfitv tal toU \01rait tw txbrby Tpbrov iftMdr, For i£ek$tai Heiberg would read {*t(tKti'w. * True, a Vatican MS. has a collection of scholia on Books 1. (extracts from the extant commentary of Proclus), It., v., vt., x. headed Eij rd EvK\tL6ov aTOiXfia TpQXtLfifitw&fitfa 4k tuv f h V''.- .V:u.-'-!T;rn,i"-a^p irai kilt' iitiTBti.-b.v* Heiberg holds that this title itself suggests that the authorship of Proclus was limited to the scholia on Book I. ; for rfnAniiparifien iit Twr llpit\ov suits extracts from Proclus' prologues, but hardly scholia to later Books. Again, a certain scholium (Heiberg in jftraun. X.XXV1II., [903, p. 341, No. 17) purports to quote words from the end of "a scholium of Proclus" on x. 9. The words quoted are from the scholium x. No. 61, one of the Scholia Vaticana. But none of the other, older, sources connect Proclus' name with X. No. 61 ; it is probable therefore that a Byiantine, who had in bis Ms, of Euclid the collection of Schol- Vat, and knew that those on !><..■!, I. came from Proclus, himself attached Proclus' name to the others- ch. iv] PROCLUS AND HIS SOURCES 33 MSS. as to suggest that the scholiast had further commentaries of Proclus which have vanished for us 1 ; (2; that there is no trace in the scholia of the notes which Proclus promised in the passages quoted above. Coming now to the question of the sources of Proclus, we may say that everything goes to show that his commentary is a compilation, though a compilation "in the better sense" of the term". He does not even give us to understand that we shall find in it much of his own ; " let us," he says, " now turn to the exposition of the theorems proved by Euclid, selecting the more subtle of the comments made on them by the ancient writers, and cutting down their interminable diffuse- ness...*": not a word about anything of his own. At the same time, he seems to imply that he will not necessarily on each occasion quote the source of each extract from an earlier commentary ; and, in fact, while he quotes the name of his authority in many places, especially where the subject is important, in many others, where it is equally certain that he is not giving anything of his own, he mentions no authority. Thus he quotes Heron by name six times ; but we now know, from the commentary of an-Nairizi, that a number of other passages, where he mentions no name, are taken from Heron, and among them the. not unimportant addition of an alternative proof to I. 19. Hence we can by no means conclude that, where no authority is mentioned, Proclus is giving notes of his own. The presumption is generally the other way ; and it is often possible to arrive at a con- clusion, either that a particular note is not Proclus' own, or that it is definitely attributable to someone else, by applying the ordinary principles of criticism. Thus, where the note shows an unmistakable affinity to another which Proclus definitely attributes to some com- mentator by name, especially when both contain some peculiar and distinctive idea, we cannot have much doubt in assigning both to the same commentator*. Again, van Pesch finds a criterion in the form of a note, where the explanation is so condensed as to be only just intelligible; the note is that in which a converse of I. 32 is proved" the proposition namely that a rectilineal figure which has all its in- terior angles together equal to two right angles is a triangle. It is not safe to attribute a passage to Proclus himself because he uses the first person in such expressions as " I say " or " I will prove " — for he was in the habit of putting into his own words the substance of notes borrowed from others — nor because, in speaking of an 1 While one class or scnuua (Schol. Vat.) have some better readings than our MSS. of Proclus h ave, and partly Fill up the gaps at 1. 36, 37 and I. 41 — 43, the other class (SchgL Vind.) derive from an inferior Proclus MS, which also had the same lacunae. a Knoche, UnUrsttchungcn iibcr dti Proklui Diattoihui Commtntar zu EttktidTi Eie- mentm {186?), p. 1 1. 3 Proclus, p. 30o, to — 13. 4 Instances of the application of this criterion will be found in the discussion of Proclus' indebtedness to the commentaries of Heron, Porphyry and Pappus, * Van Pesch attributes this converse and proof to Pappus, arguing from the fact that the proof is followed by a passage which, on comparison with Pappus' note on the postulate that all right angles are equal, he feels justified in assigning to Pappus, I doubt if the evidence is sufficient. 34 INTRODUCTION [ch. iv objection raised to a particular proposition, he uses such expressions as "perhaps someone may object" («rwe 8* &v tii>« it/trraUv...): for sometimes other words in the same passage, indicate that the objection had actually been taken by someone 1 . Speaking generally, we shall not be justified in concluding that Prod us is stating something new of his own unless he indicates this himself in express terms. As regards the form of Proclus' references to others by name, van Pesch notes that he very seldom mentions the particular zvork from which he is borrowing. If we leave out of account the references to Plato's dialogues, there are only the following references to books : the Bacchae of Philolaus 1 , the Symmikta of Porphyry*, Archimedes On the Sphere and Cylinder 1 , Apollonius On the cochlias*, a book by Eudemus on The Angle', a whole book of Posidonius directed against Zeno of the Epicurean sect', Carpus' Astronomy', Eudemus' History of Geometry*, and a tract by Ptolemy on the parallel-postulate 10 . Again, Proclus does not always indicate that he is quoting some- thing at second-hand. He often does so, e.g. he quotes Heron as the authority for a statement about Phi lip pus, Eudemus as attributing a certain theorem to Oenopides etc. ; but he says on i. 1 2 that " Oeno- pides first investigated this problem, thinking it useful for astronomy " when he cannot have had Oenopides' work before him. It has been said above that Proclus was in the habit of stating in his own words the substance of the things which he borrowed. We are prepared for this when we find him stating that he will select the best things from ancient commentaries and "cut short their intermin- able diffuseness," that he will "briefly describe" (oiWTo/ttif laTopqaai) the other proofs of 1. 20 given by Heron and Porphyry and also the proofs of I, 25 by Menelaus and Heron. But the best evidence is of course to be found in the passages where he quotes works still extant, e.g. those of Plato, Aristotle and Plotinus. Examination of these passages shows great divergences from the original; even where he purports to quote textually, using the expressions " Plato says," or " Plotinus says," he by no means quotes word for word". In fact, he seems to have had a positive distaste for quoting textually from other works. He cannot conquer this even when quoting from Euclid ; he says in his note on I. 22, " we will follow the words of the geometer " but fails, nevertheless, to reproduce the text of Euclid unchanged". We now come to the sources themselves from which Proclus drew I Van Pesch illustrates this by ;ui objection refuted in the note on I. 9, p. 1JJ, 11 sqq. After using the above expression to introduce the objection, Proclus uses further on (p. »73,ij) the term "they say" ijnelr). • Proclus, p. »a, 15. * ibid, p, j6, ij. * ibid. p. 71, 18. * ibid. p. 105, 3. ibid. p. j 35, 8. T ibid. p. aoo, 3. ■ ibid. p. j+i, 19. "' Hid. p. 351, tj. 10 ibid. p. 361, r5, II See the passages referred to by van Pesch (p. 70}. The most glaring case is a passage (p. 31, 10) where he quotes Plotinus, using the expression " Plotinus says......" Comparison with Plotinus. Hnntad. t. 3, 3, shows that very few words are those of Plotinus himself; the rest represent Plotinus' views in Proclus' own language. 11 Proclus, p. 330, 19 sqq ch. iv] PROCLUS AND HIS SOURCES 35 in writing his commentary. Three have already been disposed of, viz. Heron, Porphyry and Pappus, who had all written commentaries on the Elements 1 . We go on to Eudemus, the pupil of Aristotle, who, among other works, wrote a history of arithmetic, a history of astronomy, and a history of geometry. The importance of the last mentioned work is attested by the frequent use made of it by ancient writers. That there was no other history of geometry written after the time of Eudemus seems to be proved by the remark of Proclus in the course of his famous summary r " Those who compiled histories bring the development of this science up to this point. Not much younger than these is Euclid*. . .." The loss of Eudemus' history is one of the gravest which fate has inflicted upon us, for it cannot be doubted that Eudemus had before htm a number of the actual works of earlier geometers, which, as before observed, seem to have vanished completely when they were superseded by the treatises of Euclid, Archimedes and Apoilonius. As it is, we have to be thankful for the fragments from Eudemus which such writers as Proclus have preserved to us. I agree with van Pesch' that there is no sufficient reason for doubting that the work of Eudemus was accessible to Proclus at first hand. For the later writers Simplicius and Eutocius refer to it in terms such as leave no room for doubt that tftey had it before them. I have already quoted a passage from Simplicius' account of the lunes of Hippocrates to the effect that Eudemus must be considered the best authority since he lived nearer the times*. In the same place Simplicius says", " I will set out what Eudemus says word for word (tcwrh. \il-tv Xeyofifva), adding only a little explanation in the shape of reference to Euclid's Elements owing to the memorandum-like style of Eudemus (St a tov iriroji.viifi.aTticbv Tpoirov rov EvSjj/*ou) who sets out his explanations in the abbreviated form usual with ancient writers. Now in the second book of the history of geometry he writes as follows'." It is not possible to suppose that Simplicius would have written in this way about the style of Eudemus if he had merely been copying certain passages second-hand out of some other author and had not the original work itself to refer to. In like manner, Eutocius speaks of the paralogisms handed down in connexion with the attempts of Hippocrates and Antiphon to square; the circle", "with which I imagine that those are accurately acquainted who have examined (^reo-Ke^tei-ow) the geometrical history of Eudemus and know the Ceria Aristotelica." How could the contemporaries of Euto- cius have examined the work of Eudemus unless it was still extant in his time ? The passages in which Proclus quotes Eudemus by name as his authority are as follows : (l) On I. 26 he says that Eudemus in his history of geometry 1 See pp. 30 to a 7 above. * l*roclus, p. 68, 4—7. * De Prodi fontibui. pp. ft— Jg. * See above, p. 19. ' Simplicius, Jet. tit., ed. Dids, p. 60, 17. * Archimedes, ed. Heiberg, vol. ill. p. 11&. 36 INTRODUCTION [ch. iv referred this theorem to Thales, inasmuch as it was necessary to Thales 1 method of ascertaining the distance of ships from the shore 1 . (2) Eudemus attributed to Thales the discovery of Eucl. I. 15", and (3) to Oenopides the problem of I. 23*. (4) Eudemus referred the discovery of the theorem in 1. 32 to the Pythagoreans, and gave their proof of it, which Proclus reproduces*. (5) On I. 44 Proclus tells us* that Eudemus says that "these things are ancient, being discoveries of the Pythagorean muse, the application (irapafioXrf) of areas, their exceeding (vTrcpfioXij) and their falling short (lA-Xe^ec)," The next words about the appro- priation of these terms (parabola, hyperbola and ellipse) by later writers (i.e. Apollonius) to denote the conic sections are of course not due to Eudemus. Coming now to notes where Eudemus is not named by Proclus, we may fairly conjecture, with van Pesch, that Eudemus was really the authority for the statements (1) that Thales first proved that a circle is bisected by its diameter* (though the proof by reductio ad absurdum which follows in Proclus cannot be attributed to Thales 7 ), (z) that " Plato made over to Leodamas the analytical method, by means of which it is recorded (ltrTopi)T<u) that the latter too made many discoveries in geometry"," (3) that the theorem of I. 5 was due to Thales, and that for equal angles he used the more archaic expression "similar" angles', (4) that Oenopides first investigated the problem of I. 12, and that he called the perpendicular the gnomonic line (icara yvo>ftova.y°, (5) that the theorem that only three sorts of polygons can fill up the space round a point, viz. the equilateral triangle, the square and the regular hexagon, was Pythagorean". Eudemus may also be the authority for Proclus' description of the two methods, referred to Plato and Pythagoras respectively, of forming right-angled triangles in whole numbers". We cannot attribute to Eudemus the beginning of the note on I. 47 where Proclus says that "if we listen to those who like to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honour of his discovery"." As such a sacrifice was contrary to the Pytha- gorean tenets, and Eudemus could not have been unaware of this, the story cannot rest on his authority. Moreover Proclus speaks as though he were not certain of the correctness of the tradition ; indeed, 1 Proclus, p. 331, 14 — 18. ■ ibid, p. 199, 3. * ibid. p. 333, 5. * Hid. p. 379, 1—16. * Hid. p. + iy, ij — 18. * ibid. p. 157, 10, 11. 7 Cantor [GcscA, d. Math. Is, p. ait) points out the connexion between the rrducth ad absurdum And the Analytical method said to have been discovered by Plato, Proclus gives the proof by rfduclio ad absurdum to meet an imaginary critic who desires a mathematical proof ; possibly Thales may have been satisfied with the argument in the same sentence which mentions Thales, "the cause o r the bisection being the unswerving course of the straight line through the centre." * Proclus, p. 31 1, 19 — 13. * ibid. p. 350, 10. " ibid. p. 383, ; — 10. '* Vrid. pp. 304, 11 — 303, 3. u ibid. pp. 418, 7 — 419, 9. a Hid. p. 416, 6 — 9. ch. iv] PROCLUS AND HIS SOURCES 37 so far as the story of the sacrifice is concerned, the same thing is told of Thales in connexion with his discovery that the angle in a semi- circle is a right angle 1 , and Plutarch is not certain whether the ox was sacrificed on the discovery of I. 47 or of the problem about application of areas*. Plutarch's doubt suggests that he knew of no evidence for the story beyond the vague allusion in the distich of Apollodorus " Logisticus " (the " calculator ") cited by Diogenes Laertius also'; and Proclus may have had in mind this couplet with the passages of Plutarch. We come now to the question of the famous historical summary given by Proclus*. No one appears to maintain that Eudemus is the author of even the early part of this summary in the form in which Proclus gives it. It is, as is well known, divided into two distinct parts, between which comes the remark, " Those who compiled histories* bring the development of this science up to this point. Not much younger than these is Euclid, who put together the Elements, collecting many of the theorems of Eudoxus, perfecting many others by Theaetetus, and bringing to irrefragable demonstration the things which had only been somewhat loosely proved by his pre- decessors." Since Euclid was later than Eudemus, it is impossible that Eudemus can have written this. Yet the style of the summary after this point does not show any such change from that of the former portion as to suggest different authorship. The author of the earlier portion recurs frequently to the question of the origin of the elements of geometry in a way in which no one would be likely to do who was not later than Euclid ; and it must be the same hand which in the second portion connects Euclid's Elements with the work of Eudoxus and Theaetetus'. If then the summary is the work of one author, and that author not Eudemus, who is it likely to have been ? Tannery answers that it is Geminus 1 ; but I think, with van Pesch, that he has failed to show why it should be Geminus rather than another. And certainly the extracts which we have from Geminus' work suggest that the sort of topics which it dealt with was quite different ; they seem rather to have been general questions of the content of mathematics, and even Tannery admits that historical details could only have come inci- dentally into the work'. Could the author have been Proclus himself? Circumstances 1 Diogenes Laertius, I. 14, p. 6, ed. Cobet. 1 Plutarch, nen posse tttaviter vivi secundum Epicurum, 1 1 ; Symp- VIM, 1. ■ Ding, Laert. vm. is, p. 107, ed. Cobet: 'livipta Hv0ay6pnt ri refHK\tit fGpcTo fp&fifia, Kfii 1$ Sertfi x\uvi)v ifyaye @w0 vv^v. See on this subject Tannery, La Giomttrii gr&cqu4 t p. 105. * Proclus, pp. 64 — 70. * The plural is well explained by Tannery, La Giomitrii gricqut t pp. 73, 74. No doubt the author of the summary tried to supplement Eudemus by means of any other histories which threw light on the subject. Thus e.g. the allusion {p. 64, 11) to the Nile recalls Herodotus. Cf. the depression in Proclus, p. 64, 19, rapi Tar roXkar iirripijTiu. * Tannery, La Gimiitrit grtequt, p. ;j. ' Hid. pp. 66 — 7 j. ' ibid. p. in. 38 INTRODUCTION [ch, iv which seem to suggest this possibility are (i) that, as already stated, the question of the origin of the Elements is kept prominent, (2) that there is no mention of Democritus, whom Eudemus would not be likely to have ignored, while a fol tower of Plato would be likely enough to do him the injustice, following the example of Plato who was an opponent of Democritus, never once mentions him, and is said to have wished to burn all his writings 1 , and (3) the allusion at the beginning to the "inspired Aristotle" (0 Baifiovux; 'Apto-ToTeXi;?)*, though this may easily have been inserted by Proclus in a quotation made by him from someone else. On the other hand there are considerations which suggest that Proclus himself was not the writer. (1) The style of the whole passage is rtot such as to point to him as the author. (2) If he wrote it, it is hardly conceivable that he would have passed over in silence the discovery of the analytical method, the invention of Plato to which he attached so much importance*. There is nothing improbable in the conjecture that Proclus quoted the summary from a compendium of Eudemus' history made by some later writer: but as yet the question has not been definitely settled. All that is certain is that the early part of the summary must have been made up from scattered notices found in the great work of Eudemus. Proclus refers to another work of Eudemus besides the history, viz. a book on The Angle [0t0Xlov wepl ya>vla<;)*. Tannery assumes that this must have been part of the history, and uses this assumption to confirm his idea that the history was arranged according to subjects, not according to chronological order*. The phraseology of Proclus however unmistakably suggests a separate work ; and that the history was chronologically arranged seems to be clearly indicated by the remark of Simplicius that Eudemus "also counted Hippocrates among the more ancient writers " {iv tok vd\atoT4poi<;) a . The passage of Simplicius about the lunes of Hippocrates throws considerable light on the style of Eudemus' history. Eudemus wrote in a memorandum-like or summary manner {jov vTrofivrifiaTiKov rpoirov tow EuS?jjM>ii)' when reproducing what he found in the ancient writers; sometimes it is clear that he left out altogether proofs or constructions of things by no means easy'. Gem in u 9. The discussions about the date and birthplace of Geminus form a whole literature, as to which 1 must refer the reader to Manitius and Tittel', Though the name looks like a Latin name (Gemlnus), Mani- 1 Diog. Laertius, Ix. 40, p. 337, ltd. Cobet. - Proclus, p. 64, 8. * Proclus, p. in, [9 sqq. j the passage is quoted above, p. 36. * ibid, p. 125, 8. i Tannery, La Giotnltrit grettjite, p. 26. * Simplicius, «d. Diels, p. 60, 23. ' ibid, p. - 60, 29, * Cf. Simplicius, p. 63, 19 sqq. ; p. 64. 15 sqo. ; also Usener's note *'dc supplcndis Hippocratis quas omisit Eudemus cons tructioni bus added to Diels' preface, pp. xxiii — xxvi. 'Manitius, Gemini ticmenia. astronotnitM (Tcubner, 1898), pp. 237- — 151; Tittel, ait. *' Geminos " in Pauly-Wissowa's Rtal-Eiuytiopiidif. dtr ttassischm Aitertutnswisiemehaft, vol. Vlt.. 1910. ch. tv] PROCLUS AND HIS SOURCES 39 tius concluded that, since it appears as VeftZvos in all Greek MSS. and as rf^Ktiio? in some inscriptions, it is Greek and possibly formed from 7e/A as 'Epyiuo? is from ipy and *AX.ef wot from aXef (cf. also 'I*tm'os, KpaTivos). Tittel is equally positive that it is Gemtnus and suggests that Te/iifo? is due to a false analogy with 'AXeftetx? etc. and Fe/tetpoc wrongly formed on the model of 'Avravuvo?, ' kypwrreiva. Geminus, a Stoic philosopher, born probably in the island of Rhodes, was the author of a comprehensive work on the classification of mathematics, and also wrote, about 73-67 B.C., a not !ess comprehensive commentary on the meteorological textbook of his teacher Pnsidonius of Rhodes. It is the former work in which we are specially interested here. Though Proclus made great use of it, he does not mention its title, unless we may suppose that, in the passage (p. 177, 24) where, after quoting from Geminus a classification of lines which never meet, he says, " these remarks I have selected from the tyXoKoKia of Geminus," (ptXoxaXla is a title or an alternative title. Pappus however quotes a work of Geminus "on the classification of the mathematics" (iv t$ wept Tt}<; t&v naffyfidrtov rafetus-) 1 , while Eutocius quotes from *' the sixth book of the doctrine of the mathematics" (tv ra> eVro) 1770 t&v fiaffiHidraiv QtwpiasV Tannery' pointed out that the former title corresponds well, enough to the long extract* which Proclus gives in his first prologue, and also to the fragments contained in the Anonymi variae collectiones published by Hultsch at the end of his edition of Heron'; but it does not suit most of the o'ther passages borrowed by Proclus, The correct title was therefore probably that given by Eutocius, The Doctrine, or Theory, of the Mathematics ; and Pappus probably refers to one particular portion of the work, say the first Book. If the sixth Book treated of conies, as we may conclude from Eutocius, there must have been more Books to follow, because Proclus has preserved us details about higher curves, which must have come later. If again Geminus finished his work and wrote with the same fulness about the other branches of mathematics as he did about geometry, there must have been a considerable number of Books altogether. At all events it seems to have been designed to give a complete view of the whole science of mathematics, and in fact to be a sort of encyclopaedia of the subject. I shall now indicate first the certain, and secondly the probable, obligations of Proclus to Geminus, in which task 1 have only to follow van Pesch, who has embodied the results of Tittel's similar inquiry also*. I shall only omit the passages as regards which a case for attributing them to Geminus does not seem to me to have been made out. First come the following passages which must be attributed to Geminus, because Proclus mentions his name: (1) (In the first prologue of Proclus') on the division of mathe- 1 Pappus, ed, Hultsch, p. toi6, 9. ' Apollonius, fid. Heiberg, vol. II. p. 170. * Tannery, La Giomttru grttiiut, pp. 18, 19. * Proclus, pp. 38, I — 41, 8. * Heron, ed. Hultsch, pp. 140, 16—149, li - •Van Pesch, Di Prgck fonlibus, pp. 97—113. The dissertation of Tittel is entitled Dt Gtmini Steici ttudiit maihrmatuis ( 1895}- ' Proclus, pp, 38, I — 41, 8, except the allusion in p. 41, S — 10, to Ctesibius and Heron and 4 o INTRODUCTION [ch. iv matical sciences into arithmetic, geometry, mechanics, astronomy, optics, geodesy, canonic (science of musical harmony), and logistic (apparently arithmetical problems); (3) (in the note on the definition of a straight line) on the classification of lines (including curves) as simple (straight or circular) and mixed, composite and incomposite, uniform (ou-oiofiepels;) and non-uniform (avoitotoptpeis), lines " about solids " and lines produced by cutting solids, including conic and spiric sections' ; (3) (in the note on the definition of a plane surface) on similar distinctions extended to surfaces and solids' ; (4) (in the note on the definition of parallels) on lines which do not meet (d<rt!^wro»Toj) but which are not on that account parallel, e.g. a curve and its asymptote, showing that the property of not meeting does not make lines parallel — a favourite observation of Geminus — and, incidentally, on bounded lines or those which enclose a figure and those which do not* ; (5) (in the same note) the definition of parallels given by Posidonius' ; (6) on the distinction between postulates and axioms, the futility of trying to prove axioms, as Apollonius tried to prove Axiom 1 , and the equal incorrectness of assuming what really requires proof, " as Euclid did in the fourth postulate [equality of right angles] and in the fifth postulate [the parallel-postulate]*" ; (7) on Postulates 1, 2, 3, which Geminus makes depend on the idea of a straight line being described by the motion of a point* ; (8) (in the note on Postulate 5) on the inadmissibility in geometry of an argument which is merely plausible, and the danger in this particular case owing to the existence of lines which do converge ad infinitum and yet never meet 7 ; (9) (in the note on 1. 1) on the subject-matter of geometry, theorems, problems and Btopurftoi (conditions of possibility) for problems' ; (10) (in the note on 1, 5) on a generalisation of I. 5 by Geminus through the substitution for the rectilineal base of " one uniform line (curve)," by means of which he proved that the only " uniform lines " their pneumatic devices (fav/wrersiVinf}, as regards which ['roc! us" authority may be Pappus (ytii. p. 1014, 14 — ay) who uses very similar expressions. Heron, even if not later than Geminus, could hardly have been included in a historical work by him. Perhaps Geminus may have referred to Ctesibius only, and Proclus may have inserted " and Heron himself. 1 Proclus, pp. 103, 11 — 107, 10; pp. nt, 1 — 113, 3. * iiid. pp. 117, 14 — 110, 11, where perhaps in the pa .pp. 117, 14 — 110, it, where perhaps in the passage pp. 117, in — ti8, 1$ we may have Geminus' own words. * iiid. pp. 176, 18—177, ij; perhaps also p. 175. The note ends with the words "These things too we have selecled from Geminus' *iX«aWa for the elucidation of the matters in question." Tannery (p. 17) takes these words coming at the end of the commen- tary on the definitions as referring to the whole of the portion of Ihe commentary dealing with the definitions. Van Pesch properly regards them as only applying to the note on paralltli. This seems to me clear from the use of the word loo (TanSm mi). * Proclus, p. 176, 5 — 17. * ibid. pp. 17a — 181, 4; pp. 183, 14— 184, 10; cf. p. 188, 3— II. * ibid, p. 185, 6 — 15. ' iiid. p. 10.1, j — 19. ' ibid. pp. too, it — 101, 15. CH. iv] PROCXUS AND HIS SOURCES 41 (alike in all their parts) are a straight line, a circle, and a cylindrical helix'; (11) (in the note on I. 10) on the question whether a line is made up of indivisible parts (dfttpij), as affecting the problem of bisecting a given straight line 1 ; (12) (in the note on 1. 35) on (apical, or /o««-theorems', where the illustration of the equal parallelograms described between a hyperbola and its asymptotes may also be due to Geminus*. Other passages which may fairly be attributed to Gem in us, though his name is not mentioned, are the following : (1) in the prologue, where there is the same allusion as in the passage (8) above to a remark of Aristotle that it is equally absurd to expect scientific proofs from a rhetorician and to accept mere plausi- bilities from a geometer' ; (2) a passage in the prologue about the subject-matter, methods, and bases of geometry, the latter including axioms and postulates*; (3) another on the definition and nature of elements 1 ; (4) a remark on the Stoic use of the term axiom for every simple statement (dirotfxtvirtv <z7r\i)) s ; (5) another discussion on theorems and problems 6 , in the middle of which however there are some sentences by Prod us himself 1 '. (6) another passage, in connexion with Def. 3, on lines including or not including a figure (with which cf. part of the passage (4) above) 11 ; (7) a classification of different sorts of angles according as they are contained by simple or mixed lines (or curves)" ; (8) a similar classification of figures", and of plane figures'*; (9) Posidonius' definition of a figure 1 * ; (10) a classification of triangles into seven kinds" ; (11) a note distinguishing lines (or curves) producible indefinitely or not so producible, whether forming a figure or not forming a figure (like the "single-turn spiral") 17 ; ( 1 2) passages distinguishing different sorts of problems", different sorts of theorems", and two sorts of converses (complete and partial)*; (13) the definition of the term " porism " as used in the title of Euclid's Porisms, as distinct from the other meaning of " corollary ""; (14) a note on the Epicurean objection to I. 20 as being obvious even to an ass 11 ; (15) a passage on the properties of parallels, with allusions to I Proclus, p. aji, 1 — 11. ■ Hid. pp. 177, 15 — 179, n. * itid, pp. 394, 11— 39s, » and p. 39S, 13 — it. * Hid. p. 395, 8— 11. " »*«/- PP- 33. M— J*. I. l itid. pp, 57, 9— 58, 3. 7 itid. pp, 71, 3—75, 4. » ibid. p. 77, 3—6. ■ ibid. pp. 77, 7—78, 13, and 79, 3—81,4- l0 Hid. pp. 78, '3—79. «■ II Hid. pp. 101,-11 — 103, 18. ls Hid. pp. 116, 7 — 117, 16. » itid. pp. 159, 11— 160, 9. 14 Hid. pp. [6*, 17— 164, 6. » itid. p. 143, 5—11. " itid. p. 168, +— 11. 17 itid. p. 187, 19— 17. " Hid. pp. j jo, 7— in. 14; also p. 330, 6—9. >* itid. pp. 144, 14—1+6, 11. ' » itid. pp. 151, 5— 134, 10. " itid. pp. 301, 11—301, 13. " itid. pp. 311, 4—313. 3' 4» INTRODUCTION [ch. iv Apollonius' Conies, and the curves invented by Nicomedes, Hippias and Perseus 1 ; (16) a passage on the parallel-postulate regarded as the converse of I. i7«. Of the authors to whom P roc! us was indebted in a less degree the most important is Apollon ius of Perga. Two passages allude to his Conies*, one to a work on irrationals 1 , and two to a treatise On the cocklias (apparently the cylindrical helix) by Apollonius*. But more important for our purpose are six references to Apollonius in connexion with elementary geometry, (i) He appears as the author of an attempt to explain the idea of a line (possessing length but no breadth) by reference to daily experience, e.g. when we tell someone to measure, merely, the length of a road or of a wall'; and doubtless the similar passage showing how we may in like manner get a notion of a surface (without depth) is his also'. (2) He gave a new general definition of an angle*. (3) He tried to prove certain axioms', and Proclus gives his attempt to prove Axiom I, word for word". Proclus further quotes : (4) Apollonius' solution of the problem in Eucl. I. 10, avoiding Euclid's use of r, 9", (5) his solution of the problem in I. 11, differing only slightly from Euclid's", and (6) his solution of the problem in 1. 23 1 *, Heiberg" conjectures that Apollonius departed from Euclid's method in these propositions because he objected to solving problems of a more general, by means of problems of a more particular, character. Proclus however considers all three solutions inferior to Euclid's ; and his remarks on Apollonius' handling of these ele- mentary matters generally suggest that he was nettled by criticisms of Euclid in the work containing the things which he quotes from Apollonius, just as we conclude that Pappus was offended by the remarks of Apollonius about Euclid's incomplete treatment of the " three- and four-line locus 1 *." If this was the case, Proclus can hardly have got his information about these things at second-hand ; and there seems to be no reason to doubt that he had the actual work of Apollonius before him. This work may have been the treatise mentioned by Marinus in the words "Apollonius in his general treatise" CAirokXaivios iv r$ KaBoKov irpar/fitLTeiq) 1 '. If the notice in the Fihrist" stating, on the authority of T ha bit b. Qurra, that I Prdclus, pp. 355, 10—356, 16. ' ibid. p. 364, 9—11 ; pp. 36+, 10—365, 4. * Hid. p. 71, ig; p. 356, 8, 6. * ibid, p. 74, 13, 14. * ibid, pp. 105, s, 6, 14, 15. s Hid. p. too, 5—19. ' ibid. p. 1 [4, 10 — 13. * ibid. p. 113, I J— 19 (cf. p. 114, 17. p. t»5, ij). * ibid. p. 183, 13, 14. 10 ibid. pp. 194, 13 — J95, 5. II ibid. pp. »79, 16 — 180, 4. " ibid. p. 181, 8 — 19. 18 ibid. pp. 335, 16—336, 5. » PkiMegas, vol. xliji. p. 489. 15 See above, pp. 1, 3. '* Marinus in Euclidis Data, ed. Menge, p. 134, 16. 17 FiArist, U. Suter, p. 19. ch. iv] PROCLUS AND HIS SOURCES 43 Apollonius wrote a tract on the parallel -postulate be correct, it may have been included in the same work. We may conclude generally that, in it, Apollonius tried to remodel the beginnings of geometry, reducing the number of axioms, appealing, in his definitions of lines, surfaces etc., more to experience than to abstract reason, and substituting for certain proofs others of a more general character. The probabilities are that, in quoting from the tract of Ptolemy in which he tried to prove the para I lei -postulate, Proclus had the actual work before him. For, after an allusion to it as "a certain book 1 " he gives two long extracts', and at the beginning of the second indicates the title of the tract, "in the (book) about the meeting of straight lines produced from (angles) less than two right angles," as he has very rarely done in other cases. Certain things from Posidonius are evidently quoted at second- hand, the authority being Geminus (e.g. the definitions of figure and parallels) ; but besides these we have quotations from a separate work which he wrote to controvert Zeno of Sidon, an Epicurean who had sought to destroy the whole of geometry*. We are told that Ze.10 had argued that, even if we admit the fundamental principles (tipx<U) of geometry, the deductions from them cannot be proved without the admission of something else as well, which has not been included in the said principles*. On I. 1 Proclus gives at some length the argu- ments of Zeno and the reply of Posidonius as regards this proposition'. In this case Zeno's "something else" which he considers to be assumed is the fact that two straight lines cannot have a common segment, and then, as regards the " proof " of it by means of the bisection of a circle by its diameter, he objects that it has been assumed that two circumferences (arcs) of circles cannot have a common part. Lastly, he makes up, for the purpose of attacking it, another supposed " proof" of the fact that two straight lines cannot have a common part. Proclus appears, more than once, to be quoting the actual words of Zeno and Posidonius ; in particular, two expres- sions used by Posidonius about " the acrid Epicurean " (tov bptfivv 'Eirucovpciav)* and his "misrepresentations" (Ho<ret8<bvi6<; ty)<rf rbv Ztjcowa <ruKo<f>avT€lvY. It is not necessary to suppose that Proclus had the original work of Zeno before him, because Zeno's arguments may easily have been got from Posidonius' reply ; but he would appear to have quoted direct from the latter at all events. The work of Carpus tnec/tankus (a treatise on astronomy) quoted from by Proclus" must have been accessible to him at first-hand, because a portion of the extract from it about the relation of theorems and problems" is reproduced word for word. Moreover, if he were not using the book itself, Proclus would hardly be in a position to question whether the introduction of the subject of theorems and problems ' Proclus, p. 191, 13. ' ibid. pp. 363, 14— 363, IB; pp. 36s, 7—3671 *?■ ' ibid, p. loo, 1—3. * ibid. pp. 199, u— 200, 1. * ibid. pp. »i4, [8 — iif, 13 [ pp. 516, 10 — 118, M. 4 ibid. p. 3l6, 11. 7 ibid. p. ?]8, I. * ibid, pp. J41, 19 — »43, 11. ' Md, pp. 141, n — 143, 11. 44 INTRODUCTION [ch. iv was opportune in the place where it was I'ound («* piv Kara Kaipiv rj ^»i, Trapela0a> wpos to irapov) 1 . It is of course evident that Proclus had before him the original works of Piato, Aristotle, Archimedes and Plotinus, as well as the Ivfifiutrd of Porphyry and the works of his master Syrianus (d ^ftirepo^ Ka8fjyefta)v) % , from whom he quotes in his note on the definition of an angle. Tannery also points out that he must have had before him a group of works representing the Pythagorean tradition on its mystic, as distinct from its mathematical, side, from Philolaus downwards, and comprising the more or less apocryphal Upfc Xoyai of Pythagoras, the Oracles (\6yta), and Orphic verses'. Besides quotations from writers whom we can identify with more or less certainty, there are many other passages which are doubtless quoted from other commentators whose names we do not know. A list of such passages is given by van Pesch*, and there is no need to cite them here. Van Pesch also gives at the end of his work" a convenient list of the books which, as the result of his investigation, he deems to have been accessible to and directly used by Proclus, The list is worth giving here, on the same ground of convenience. It is as follows: Eudemus : history of geometry. Gem in us : the theory of the mathematical sciences. Heron : commentary on the Elements of Euclid. Porphyry: „ „ Pappus t „ „ „ Apollonius of Perga : a work relating to elementary geometry. Ptolemy : on the parallel-postulate. Posidonius : a book controverting Zeno of Si don. Carpus : astronomy. Syrianus j a discussion on the angle. Pythagorean philosophical tradition. Plato's works. Aristotle's works. Archimedes' works, Plotinus : Enneaaes. Lastly we come to the question what passages, if any, in the commentary of Proclus represent his own contributions to the subject As we have seen, the onus probandi must be held to rest upon him who shall maintain that a particular note is original on the part of Proclus, Hence it is not enough that it should be impossible to point to another writer as the probable source of a note ; we must have a positive reason for attributing it to Proclus, The criterion must there- fore be found either (i) in the general terms in which Proclus points out the deficiencies in previous commentaries and indicates the respects in which his own will differ from them, or (2) in specific expressions used by him in introducing particular notes which may 1 Proclus, p. 741, j 1, 11. ■ ibid. p. 113, 19, 8 Tannery, La GionUtrie grecque, pp. 15, 16, * Van Pesch, Dt Prvcli fetttitrus, p. 135. * ibid. p. 155, CH. iv] PROCLUS AND HIS SOURCES 45 indicate that he is giving his own views. Besides indicating that he paid more attention than his predecessors to questions requiring deeper study (to irpayuarettSSes) and " pursued clear distinctions ' {to evStaiperav (ieTa8ia>Kovra<:) x — by which he appears to imply that his predecessors had confused the different departments of their commentaries, viz. lemmas, cases, and objections (eyo-rrwret?)* — Proclus complains that the earlier commentators had failed to indicate the ultimate grounds or muses of propositions*. Although it is from Geminus that he borrowed a passage maintaining that it is one of the proper functions of geometry to inquire into causes (tijii alrtav koX to S*a Tt)*, yet it is not likely that Geminus dealt with Euclid's propositions one by one ; and consequently, when we find Proclus, on I. 8, 16, 17, 18, 32, and 47', endeavouring to explain causes, we have good reason to suppose that the explanations are his own. Again, his remarks on certain things which he quotes from Pappus can scarcely be due to anyone else, since Pappus is the latest of the commentators whose works he appears to have used. Under this head, come (1) his objections to certain new axioms introduced by Pappus*, (2) his conjecture as to how Pappus came to think of his alterna- tive proof of I. 5', (3) an addition to Pappus' remarks about the curvilineal angle which is equal to a right angle without being one*. The defence of Geminus against Carpus, who combated his view of theorems and problems, is also probably due to Proclus', as well as an observation on t. 38 to the effect that I. 35 — 38 are really compre- hended in VI. 1 as particular cases". Lastly, we can have no hesitation in attributing to Proclus himself (1) the criticism of Ptolemy's attempt to prove the parallel-postulate", and (2) the other attempted proof given in the same note" (on I. 29) and assuming as an axiom that " if from one point two straight lines forming an angle be produced ad infinitum the distance between them when so produced ad infinitum exceeds any finite magnitude (i.e. length)," an assumption which purports to be the equivalent of a statement in Aristotle 1 *. It is introduced by words in which the writer appears to claim originality for his proof: "To him who desires to see this proved (Ka-Ta&Kevatyfieiiov) let it be said by us (Xeyia-ff& trap thiwv)" etc." Moreover, Philoponus, in a note on Aristotle's Anal. pest. 1. to, says that " the geometer (Euclid) assumes this as an axiom, but it wants a great deal of proof, insomuch that both Ptolemy and Proclus wrote a whole book upon it"." 1 Proclui, p. 84, 1 j, p. 431, ff, j j. ■ Cf. ibid, p. *8g, u— ij ; p. 431, 13— ij. ibid. p. 431, 17. * ibid, p. 103, y—^$. Proclus, p. II pp. 316, T+- _ Proclus, p. 198, 5 — 15. 1 ibid, p. 150, 11—19. * ibid. p. 190, 9 — ^13. * See Proclus, p. 170, 3—14 (1. 8); pp. 309, 3—310, 8 (1. 16); pp. 310, 19—311, 33 •T)( PP' 3'<>. '+—318, » ('• '8); p. 384, 13— j 1 (1. 31) s pp. +16, 11— 417. 8 ft. 47). * ibid, p. 343, 11 — 19. * ibid. pp. 403, 6—406, 9. 11 ibid. p. 368, 1— 13. u ibid. pp. 371, 11—373, »■ u Aristotle, de caila, 1. 3 (171 b 18 — 30), " Produs, p. 371, 10. u Berlin Aristotle, voL IV. p. 114 a 9 — 11. CHAPTER V. THE TEXT 1 . It is well known that the title of Simson's edition of Euclid (first brought out in 'Latin and English in 1756) claims that, in it, "the errors by which Theon, or others, have long ago vitiated these books are corrected, and some of Euclid's demonstrations are restored " ; and readers of Simson's notes are familiar with the phrases used, where anything in the text does not seem to him satisfactory, to the effect that the demonstration has been spoiled, or things have been interpo- lated or omitted, by Theon "or some other unskilful editor." Now most of the MSS. of the Greek text prove by their titles that they proceed from the recension of the Elements by Theon ; they purport to be either " from the edition of Theon "(«tij? ©eWoe e'wSoVew?) or " from the lectures of Theon " (Avo avvovaw&v tov QsWoc). This was Theon of Alexandria {4th c. A.D.) who also wrote a commentary on Ptolemy, in which there occurs a passage of the greatest importance in this connexion*: "But that sectors in equal circles are to one another as the angles on which they stand has been proved by me in my edition of the Elements at the end of the sixth book." Thus Theon himself says that he edited the Elements and also that the second part of VI. 33, found in nearly all the MSS., is his addition. This passage is the key to the whole question of Th eon's changes in the text of Euclid ; for, when Peyrard found in the Vatican the MS. 190 which contained neither the words from the titles of the other MSS. quoted above nor the interpolated second part of VI. 33, he was justified in concluding, as he did, that in the Vatican MS. we have an edition more ancient than Theon 's. It is also clear that the copyist of P, or rather of its archetype, had before him the two recensions and systematically gave the preference to the earlier one ; for at xiii. 6 in P the first hand has added a note in the margin : " This theorem is not given in most copies of the new edition, but is found in those of the old." Thus we are more fortunate than Simson, since our judgment of Theon's recension can be formed on the basis, not of mere conjecture, but of the documentary evidence afforded by a comparison of the Vatican MS. just mentioned with what we may conveniently call, after Heiberg, the Theonine MSS. 1 The material for the whole of this chapter is taken from Hei berg's edition of the Elements, introduction to vol. v., and from the same scholar's Littcrargcscktihlltihc Siudien iibcr Euilid, p. tJ4sqq. and ParaHpomena zit EuMidm Her-mcs, XXXVltt., too}. 9 1. p. J 01 ed. Ililmj = |i. 50 ed. Basel. ch. v] THE TEXT 47 The MSS. used for Hei berg's edition of the Elements are the following : (i) P = Vatican MS. numbered 190, 4to, in two volumes (doubt- less one originally) ; 10th c. This is the MS. which Peyrard was able to use ; it was sent from Rome to Paris for his use and bears the stamp of the Paris Imperial Library on the last page. It is well and carefully written. There are corrections some of which are by the original hand, but generally in paler ink, others, still pretty old, by several different hands, or by one hand with different ink in different places (P m. 2), and others again by the latest hand (P m. rec). It contains, first, the Elements I. — xin. with scholia, then Marinus' commentary on the Data (without the name of the author), followed by the Data itself and scholia, then the Elements XIV., XV. (so called), and lastly three books and a part of a fourth of a commentary by Theon eis rot)* irpoxeipow Kavovat IlToXe- fiaiou. The other MSS. are " Theon ine." (2) F = MS. xxvili, 3, in the Laurentian Library at Florence, 4to; 10th c. This MS. is written in a beautiful and scholarly hand and contains the Elements I. — XV., the Optics and the Phaenotnena, but is not well preserved. Not only is the original writing renewed in many places, where it had become faint, by a later hand of the 1 6th c, but the same hand has filled certain smaller lacunae by gumming on to torn pages new pieces of parchment, and has replaced bodily certain portions of the MS., which had doubtless become illegible, by fresh leaves. The larger gaps so made good extend from Eucl. Vlt. 1 2 to IX. 1 5, and from xil. 3 to the end ; so that, besides the conclusion of the Elements, the Optics and Phaenomena are also in the later hand, and we cannot even tell what in addition to the Elements 1. — xin. the original MS. contained. Hei berg denotes the later hand by <f> and observes that, while in restoring wo;ds which had become faint and filling up minor lacunae the writer used no other MS., yet in the two larger restorations he used the Laurentian MS. XXVIII, 6, belonging to the 13th — 14th c. The latter MS. (which Heiberg denotes by f) was copied from the Viennese MS. (V) to be described below. (3) B m Bodleian MS., D'Orville X. 1 inf. 2, 30, 4to ; A.D. 888. This MS. contains the Elements I. — XV. with many scholia. Leaves 15 — 118 contain 1. 14 (from about the middle of the proposition) to the end of Book VI., and leaves 123 — 387 (wrongly numbered 397) Books VII, — XV, in one and the same elegant hand (9th c). The leaves preceding leaf 15 seem to have been lost at some time, leaves 6 to 14 (containing Elem. L to the place in I. 14 above referred to) being carelessly written by a later hand on thick and common parch- ment (13th c). On leaves 2 to 4 and 122 are certain notes in the hand of Arethas, who also wrote a two-line epigram on leaf 5, the greater part of the scholia in uncial letters, a few notes and corrections, and two sentences on the last leaf, the first of which states that the MS. was written by one Stephen cleric us in the year of the world 6397 48 INTRODUCTION [ch. v (= 888 a.D,), while the second records Arethas' own acquisition of it. Arethas lived from, say, 865 to 939 a.D. He was Archbishop of Caesarea and wrote a commentary on the Apocalypse. The portions of his library which survive are of the greatest interest to palaeography on account of his exact notes of dates, names of copyists, prices of parchment etc. It is to him also that we owe the famous Plato MS. from Fatmos (Cod. Clarkianus) which was written for him in November 895 ! . (4) V - Viennese MS. Philos. Gr. No. 103 ; probably 12th c. This MS. contains 292 leaves, Eucl, Elements I. — XV. occupying leaves 1 to 254, after which come the Optics (to leaf 271), the P%aenomena (mutilated at the end) from leaf 372 to leaf 282, and lastly scholia, on leaves 283 to 292, also imperfect at the end. The different material used for different parts and the varieties of handwriting make it necessary for Heiberg to discuss this ms. at some length'. The handwriting on leaves 1 to 183 (Book 1. to the middle of X. 105) and on leaves 203 to 234 (from XI 31, towards the end of the proposition, to XIII. 7, a few lines down) is the same; between leaves 184 and 202 there are two varieties of handwriting, that of leaves 184 to 189 and that of leaves 200 (verso) to 202 being the same. Leaf 235 begins in the same handwriting, changes first gradually into that of leaves 184 to 189 and then (verso) into a third more rapid cursive writing which is the same as that of the greater part of the scholia, and also as that of leaves 243 and 282, although, as these leaves are of different material, the look of the writing and of the ink seems altered. There are corrections both by the first and a second hand, and scholia by many hands. On the whole, in spite of the apparent diversity of handwriting in the MS., it is probable that the whole of it was written at about the same time, and it may (allowing for changes of material, ink etc) even have been written by the same man. It is at least certain that, when the Laurentian ms. xxviii, 6 was copied from it, the whole MS, was in the condition in which it is now, except as regards the later scholia and leaves 283 to 292 which are not in the Laurentian MS., that MS. coming to an end where the Phaenomena breaks off abruptly in V, Hence Heiberg attributes the whole MS. to the izthc. But it was apparently in two volumes originally, the first con- sisting of leaves 1 to 183 ; and it is certain that it was not all copied at the same time or from one and the same original. For leaves 1 84 to 202 were evidently copied from two MSS, different both from one another and from that from which the rest was copied. Leaves 184 to the middle of leaf 189 (recto) must have been copied from a MS. similar to P, as is proved by similarity of readings, though not from F itself. The rest, up to leaf 202, were copied from the Bologna MS. (b) to be mentioned below. It seems clear that the content of leaves 1 84 to 202 was supplied from other MSS. because there was a lacuna in the original from which the rest of V was copied. 1 See Pauty-Wissowa, Ktal- Encyclopedic dtr class. AUertitiHSWitunickaft, vol. [I., 1896, dberg, vol. v. pp. xxts — xxxiii. * **fc ch. v] THE TEXT 49 Heiberg sums up his conclusions thus. The copyist of V first copied leaves i to 183 from an original in which two quaterntones were missing (covering from the middle of Eucl. X. 105 to near the end of XI. 31). Noticing the lacuna he put aside one quatemio of the parchment used up to that point. Then he copied onwards from the end of the lacuna in the original to the end of the Pkaenomena. After this he looked about him for another ms. from which to fill up the lacuna; finding one, he copied from it as far as the middle of leaf 189 (recto). Then, noticing that the MS. from which he was copying was of a different class, he had recourse to yet another MS. from which he copied up to leaf 302, At the same time, finding that the lacuna was longer than he had reckoned for, he had to use twelve more leaves of a different parchment in addition to the quatemio which he had put aside. The whole MS. at first formed two volumes (the first containing leaves 1 to 1 83 and the second leaves 1 84 to 282) ; then, after the last leaf had perished, the two volumes were made into one to which two more quaterniones were also added. A few leaves of the latter of these two have since perished, (5) b = MS. numbered 18 — 19 in the Communal Library at Bologna, in two volumes, 4to ; nth c. This MS. has scholia in the margin written both by the first hand and by two or three later hands ; some are written by the latest hand, Theodorus Cabasilas (a descendant apparently of Nicolaus Cabasilas, 14th c.) who owned the MS. at one time. It contains (a) in 14 quater- niones the definitions and the enunciations (without proofs) of the Elements I. — XIII, and of the Data, {b) in the remainder of the volumes the Proem to Geometry (published among the Variae Collectiones in Hultsch's edition of Heron, pp. 252, 24 to 274, 14) followed by the Elements I. — XIII. (part of XIII. 18 to the end being missing), and then by part of the Data (from the last three words of the enunciation of Prop. 38 to the end of the penultimate clause in Prop. 87, ed. Menge). From xi. 36 inclusive to the end of xn. this MS. appears to represent an entirely different recension. Heiberg is compelled to give this portion of b separately in an appendix. He conjectures that it is due to a Byzantine mathematician who thought Euclid's proofs too long and tiresome and consequently contented himself with indicating the course followed 1 . At the same time this Byzantine must have had an excellent MS. before him, probably of the ante-Theonine variety of which the Vatican Ms. 190 (P) is the sole representative. (6) p = Paris MS. 2466, 41.0; 12th c This manuscript is written in two hands, the finer hand occupying leaves 1 to 53 (recto), and a more careless hand leaves 53 (verso) to 64, which are of the same parchment as the earlier leaves, and leaves 65 to 239, which are of a thinner and rougher parchment showing traces of writing of the 8th — 9th c (a Greek version of the Old Testament). The ms, contains the Elements 1. — xm. and some scholia after Books XL, XII. and xm. 1 ZtUs<hriflfur Math, u. Pkysik, xxix., hiit.-litt. Abtheilung, p. [3. 50 INTRODUCTION [ch, v (7) 1 ~ Paris MS. 2344, folio ; 12th c. It is written by one hand but includes scholia by many hands. On leaves 1 to 16 (recto) are scholia with the same title as that found by Wachsmuth in a Vatican MS. and relied upon by him to prove that Proclus continued his commentaries beyond Book I. 1 Leaves 17 to 357 contain the Elements I. — Xltl. (except that there is a lacuna from the middle of VIII. 25 to the «&<rn? of ix. 14) ; before Books VII. and X. there are some leaves filled with scholia only, and leaves 358 to 366 contain nothing but scholia. (8) Heiberg also used a palimpsest in the British Museum (Add. 17211). Five pages are of the 7th— 8th c. and are contained (leaves 49 — 53) in the second volume of the Syrian MS. Brit. Mus. 687 of the 9th a ; half of leaf 50 has perished. The leaves contain various frag- ments from Book x, enumerated by Heiberg, Vol. HI., p. v, and nearly the whole of XIII. 14. Since his edition of the Elements was published, Heiberg has collected further material bearing on the history of the text". Besides giving the results of further or new examination of MSS.. he has collected the fresh evidence contained in an-Nairizi's commentary, and particularly in the quotations from Heron's commentary given in it (often word for word), which enable us in several cases to trace differences between our text and the text as Heron had it, and to identify some interpolations which actually found their way into the text from Heron's commentary itself; and lastly he has dealt with some valuable fragments of ancient papyri which have recently come to light, and which are especially important in that the evidence drawn from them necessitates some modification in the views expressed in the preface to Vol. V. as to the nature of the changes made in Theon's recension, and in the principles laid down for differentiating between Theon's recension and the original text, on the basis of a comparison between P and the Theonine MSS. alone. The fragments of ancient papyri referred to are the following. 1. Papyrus Herculanensis No. 1061'. This fragment quotes Def. 15 of Book I. in Greek, and omits the words jj icaXelrai irepupepem, "which is called the circumference," found in all our MSS., and the further addition Trpbt t>)v toD kvk\ov V€ptif>epeiav also found in practically all the MSS. Thus Heiberg's assumption that both expressions are interpolations is now conSrmed by this oldest of all sources. 2. The Oxyrhynchus Papyri \. p. 58, No. xxix. of the 3rd or 4U1 c. This fragment contains the enunciation of Eucl. II. 5 (with figure, apparently without letters, immediately following, and not, as usual in our MSS., at the end of the proof) and before it the part of a word irept#xpp£ belonging to II. 4 (with room for -v<p 6p8oyaiviq>' cmep ehu 1 [dt r]A rod EtixXtloav rrotxtia Tpo\at*fia*6tti*& i* w6r Hp&cXou wopdiijv *al rar' /rt- rdp^r. Cf. p. 33, note 8, Above. ■ Heiberg, Paralipemtna ;u liuklid in Htrma, XXXV MI., 190 J, pp. 46 — 74, ifil — aoi, 311—356. 1 Described by Heiberg in Overrigt ever dtt kngi, dansks Vtdtnskabtrittt Sthkabs Forhandiingtr, rooo, p. 161. em v] THE TEXT 51 Sei|o( and a stroke to mark the end), showing that the fragment had not the Porism which appears in all the Theonine MSS. and (in a later hand) in P, and thereby confirming Hetberg's assumption that the Porism was due to Theon. 3, A fragment in Fayum towns and their papyri, p. 96, No. IX. of 2nd or 3rd c. This contains I. 39 and I. 41 following one another and almost complete, showing that I. 40 was wanting, whereas it is found in all the MSS. and is recognised by Prod us. Moreover the text of the beginning of I. 39 is better than ours, since it has no double Biopwfios but omits the first (" I say that they are also in the same parallels ") and has " and" instead of "for let AD be joined " in the next sentence. It is clear that I. 40 was interpolated by someone who thought there ought to be a proposition following I. 39 and related to it as I. 38 is related to I. 37 and I. 36 to I. 35, although Euclid nowhere uses 1. 40, and therefore was not likely to include it The same interpolator failed to realise that the words " let AD be joined" were part of the eK^ca-is or setting-out, and took them for the *oTocr*einj or " construc- tion " which generally follows the 8(op«r^>? or " particular statement " of the conclusion to be proved, and consequently thought it necessary to insert a &topurfi,6<! before the words. The conclusions drawn by Heiberg from a consideration of particular readings in this papyrus along with those of our MSS. will be referred to below. We now come to the principles which Heiberg followed, when preparing his edition, in differentiating the original text from the Theonine recension by means of a comparison of the readings of P and of the Theonine MSS. The rules which he gives are subject to a certain number of exceptions (mostly in cases where one Ms. or the other shows readings due to copyists' errors), but in general they may be relied upon to give conclusive results. The possible alternatives which the comparison of P with the Theonine mss. may give in particular passages are as follows : I. There may be agreement in three different degrees. (1) P and all the Theonine MSS. may agree. In this case the reading common to all, even if it is corrupt or interpolated, is more ancient than Theon, i.e. than the 4th c. (2) P may agree with some (only) of the Theonine MSS. In this case Heiberg considered that the latter give the true reading of Theon's recension, and the other Theonine MSS. have departed from it (3) P and one only of the Theonine MSS may agree. In this case too Heiberg assumed that the one Theonine MS. which agrees with P gives the true Theonine reading, and that this rule even supplies a sort of measure of the quality and faithfulness of the Theonine MSS. Now none of them agrees alone \.ith P in preserving the true reading so often as F. Hence F must be held to havt pre- served Theon's recension more faithfully than the other Theonine MSS. ; and it would follow that in those portions where F fails us P must 5» INTRODUCTION [en. v carry rather more weight even though it may differ from the Theortine MSS. BVpq. (Heiberg gives many examples in proof of this, as of his main rules generally, for which reference must be made to his Prole- gomena in Vol. V.) The specially close relation of F and P is also illustrated by passages in which they have the same errors; the explanation of these common errors (where not due to accident) is found by Heiberg in the supposition that they existed, but were not noticed by Theon, in the original copy in which he made his changes. Although however F is by far the best of the Theonine MSS,, there are a considerable number of passages where one of the others (B, V, p or q) alone with P gives the genuine reading of Theon 's recension. As the result of the discovery of the papyrus fragment containing l. 39, 41, the principles above enunciated under (2) and (3) are found by Heiberg to require some qualification. For there is in some cases a remarkable agreement between the papyrus and the Theonine MSS. (some or all) as against P. This shows that Theon took more trouble to follow older MSS., and made fewer arbitrary changes of his own, than has hitherto been supposed. Next, when the papyrus agrees with some of the Theonine MSS. against P, it must now be held that these mss, (and not, as formerly supposed, those which agree with P) give the true reading of Theon. If it were otherwise, the agreement between the papyrus and the Theonine MSS. would be accidental: but it happens too often for this. It is clear also that there must have been contamination between the two recensions ; otherwise, whence could the Theonine mss. which agree with P and not with the papyrus have got their readings? The influence of the P class on the Theonine F is especially marked. II. There may be disagreement between P and all the Theonine MSS. The following possibilities arise, (1) The Theonine MSS. differ also among themselves. In this case Heiberg considered that P nearly always has the true reading, and the Theonine MSS. have suffered interpolation in different ways after Theon's time. (2) The Theonine MSS. all combine against P. In this case the explanation was assumed by Heiberg to be one or other of the following. (<*) The common reading is due to an error which cannot be imputed to Theon (though it may have escaped him when putting together the archetype of his edition); such error may either have arisen accidentally in all alike, or (more frequently) may be referred to a common archetype of all the MSS. (fi) There may be an accidental error in P ; e.g. something has dropped out of P in a good many places, generally through 6fiOlOT&€VTOV (<y) There may be words interpolated in P. (0) Lastly, we may have in the Theonine MSS. a change made by Theon himself. (The discovery of the ancient papyrus showing readings agreeing ch. v] THE TEXT 5S with some, or with all, of the Theonine MSS. against P now makes tt necessary to be very cautious in applying these criteria.) It is of course the last class (8) of changes which we have to investigate in order to get a proper idea of Theon's recension. Heiberg first observes, as regards these, that we shall find that Theon, in editing the Elements, altered hardly anything without some reason, often inadequate according to our ideas, but still some reason which seemed to htm sufficient Hence, in cases of very slight differ- ences where both the Theonine MSS. and P have readings good and probable in themselves, Heiberg is not prepared to put the differences down to Theon. In those passages where we cannot see the least reason why Theon, if he had the reading of P before him, should have altered it, Heiberg would not at once assume the superiority of P unless there was such a consistency in the differences as wouW indicate that they were due not to accident but to design. In the absence of such indications, he thinks that the ordinary principles of criticism should be followed and that proper weight should be attached to the antiquity of the sources. And it cannot be denied that the sources of the Theonine version are the more ancient. For not only is the British Museum palimpsest (L), which is intimately connected with the rest of our MSS., at least two centuries older than P, but the other Theonine M5S. are so nearly allied that they must be held to have had a common archetype intermediate between them and the actual edition of Theon ; and, since they themselves are as old as, or older than P, their archetype must have been much older. Heiberg gives Spp. xlvi, xlvii) a list of passages where, for this reason, he has bl lowed the Theonine MSS. in preference to P, It has been mentioned above that the copyist of P or rather of its archetype wished to give an ancient recension. Therefore (apart from clerical errors and interpolations) the first hand in P may be relied upon as giving a genuine reading even where a correction by the first hand has been made at the same time. But in many places the first hand has made corrections afterwards ; on these occasions he must have used new sources, e.g. when inserting the scholia to the first Book which P alone has, and in a number of passages he has made additions from Theonine MSS. We cannot make out any " family tree " for the different Theonine MSS. Although they all proceeded from a common archetype later than the edition of Theon itself, they cannot have been copied one from the other ; for, if they har' been, how could it have come about that in one place or other each of them agrees alette with P in pre- serving the genuine reading .' Moreover the great variety in their agreements and disagreements indicates that they have all diverged to about the same extent from their archetype. As we have seen that P contains corrections from the Theonine family, so they show correc- tions from P or other MSS. of the same family. Thus V has part of the lacuna in the ms. from which it was copied filled up from a MS. similar to P, and has corrections apparently derived from the same ; the copyist, however, in correcting V, also used another MS. to which 54 INTRODUCTION [ch. v he alludes in the additions to IX. 19 and 30 (and also on x. 23 Por.): "in the book of the Ephesian (this) is not found," Who this Ephesian of the 12th c. was, we do not know. We now come to the alterations made by Theon in his edition of the Elements. I shall indicate classes into which these alterations may be divided but without details (except in cases where they affect the mathematical content as distinct from form or language pure and simple) 1 . I. Alterations made by Theon where he found, or thought he found, mistakes in the original. 1. Real blots in the original which Theon saw and tried to remove. (a) Euclid has a porism (corollary) to VI. 19, the enunciation of which speaks of similar and similarly described figures though the proposition itself refers only to triangles, and therefore the porism should have come after VI. 20. Theon substitutes triangle for figure and proves the more general porism after vi. 2a (o) In IX. 19 there is a statement which is obviously incorrect. Theon saw this and altered the proof by reducing four alternatives to two, with the result that it fails to correspond to the enunciation even with Theon 's substitution of "if" for "when" in the enunciation. (c) Theon omits a porism to IX. 11, although it is necessary for the proof of the succeeding proposition, apparently because, owing to an error in the text (kcitcL tov corrected by Heiberg into «ri to), he could not get out of it the right sense. (d) I should also put into this category a case which Heiberg classifies among those in which Theon merely fancied that he found mistakes, viz. the porism to V. 7 stating that, if four magnitudes are proportional, they are proportional inversely. Theon puts this after v, 4 with a proof, which however has no necessary connexion with V. 4 but is obvious from the definition of proportion, (e) I should also put under this head XI. I, where Euclid's argu- ment to prove that two straight lines cannot have a common segment is altered. 2. Passages which seemed to Theon to contain blots, and which he therefore set himself to correct, though more careful consideration would have shown that Euclid's words are right or at least may be excused and offer no difficulty to an intelligent reader. Under this head come : (a) an alteration in III. 24. {b) a perfectly unnecessary alteration, in VI, 14, of " equiangular parallelograms " into " parallelograms having one angle equal to one angle," where Theon followed the false analogy of VI. 15. (c) an omission of words in V. 26, owing to his having been mis- led by a wrong figure. (d) an alteration of the order of xi. Deff. 27, 28. (e) the substitution of " parallelepiped a I solid " for " cube " in XI. 1 Exhaustive details under all the different heads are given by Heiberg (Vol. v. pp. lii — Uxv). ch. v] THE TEXT $S 38, because Theon observed, correctly enough, that it was true of the parallelepipedal solid in general as well as of the cube, but failed to give weight to the fact that Euclid must have given the particular case of the cube for the simple reason that that was all he wanted for use in XIII, 17. {/) the substitution of the letter 4> for fl ( V for Z in my figure) because he saw that the perpendicular from K to B4> would fall on <£ itself, so that <t>, tl coincide. But, if the substitution is made, it should be proved that 4>, £1 coincide. Euclid can hardly have failed to notice the fact, but it may be that he deliberately ignored it as unnecessary for his purpose, because he did not .want to lengthen his proposition by giving the proof. I I. Emendations intended to improve the form or diction of Euclid. Some of these emendations of Theon affect passages of appreciable length. Heiberg notes about ten such passages ; the longest is in Eucl. xif. 4 where a whole page of Heiberg's text is affected and Theon's version is put in the Appendix. The kind of alteration may be illustrated by that in ix. r 5 where Euclid uses successively the propositions VII, 24, 25, quoting the enunciation of the former but not of the latter ; Theon does exactly the reverse. In a few of the cases here quoted by Heiberg, Theon shortened the original somewhat. But, as a rule, the emendations affect only a few words in each sentence. Sometimes they are considerable enough to alter the con- formation of the sentence, sometimes they are trifling alterations "more magistellorum ineptorum" and unworthy of Theon. Generally speaking, they were prompted by a desire to change anything which was out of the common in expression or in form, in order to reduce the language to one and the same standard or norm. Thus Theon changed the order of words, substituted one word for another where the latter was used in a sense unusual with Euclid (e.g. i-ireihijirep, " since," for on. in the sense of " because "), or one expression for another in like circumstances (e.g. where, finding "that which was enjoined would be done" in a theorem, VII. 31, and deeming the phrase more appropriate to a problem, he substituted for it " that which is sought would be manifest"; probably also and for similar reasons he made certain variations between the two expressions usual at the end of propositions otrep eSei Setl-ai and oirep (Bet iroifjerat, quod erat demonstrandum and quod erat faciendum). Sometimes his alterations show carelessness in the use of technical terms, as when he uses aTTTtaSai (to meet) for t4>d-n-T£<r$at, (to touch) although the ancients carefully distinguished the two words. The desire of keeping to a standard phraseology also led Theon to omit or add words in a number of cases, and also, sometimes, to change the lettering of figures. But Theon seems, in editing the Elements, to have bestowed the most attention upon III. Additions designed to supplement or explain Euclid. First, he did not hesitate to interpolate whole propositions where he thought there was room or use for them. We have already 56 INTRODUCTION [ch. v mentioned the addition to VI. 33 of the second part relating to sectors, for which Theon himself takes credit in his commentary on Ptolemy. Again, he interpolated the proposition commonly known as vn. 22 (ex aequo in proportions perturbata for numbers, corresponding to V. 23), and perhaps also vn. 20, a particular case of VII. 19 as VI. 17 is of VI. 16. He added a second case to VI. 27, a porism to II. 4, a second porism to III. 1 6, and a lemma after X. 12 ; perhaps also the porism to V. 19 and the first porism to vi. 2a He also inserted alternative proofs here and there, e.g. in II. 4 (where the alternative differs little from the original) and in vn. 3 1 ; perhaps also in X. 1 , 6, and 9. Secondly, he sometimes repeats an argument where Euclid had said " For the same reason," adds specific references to points, straight lines etc. in the figures in order to exclude the possibility of mistake arising from Euclid's reference to them in general terms, or inserts words to make the meaning of Euclid more plain, e.g. componendo and alternately, where Euclid had left them out. Some- times he thought to increase by his additions the mathematical precision of Euclid's language in enunciations or elsewhere, sometimes to make smoother and clearer things which Euclid had expressed with unusual brevity and harshness or carelessness, in reliance on the intelligence of his readers. Thirdly, he supplied intermediate steps where Euclid's argument seemed too rapid and not easy enough to follow. The form of these additions varies ; they are sometimes placed as a definite intermediate step with " therefore " or " so that," sometimes they are additions to the statement of premisses, sometimes phrases introduced by "since," "for" and the like, after the inference. Lastly, there is a very large class of additions of a word, or one or two words, for the sake of clearness or consistency. Heiberg gives a number of examples of the addition of such nouns as "triangle," "square," "rectangle," "magnitude," "number," "point," "side," "circle," "straight line," "area" and the like, of adjectives such as " remaining," " right," " whole," " proportional," and of other parts of speech, even down to words like " is " (4(ttC) which is added 600 times, Si;, upa, ftiv, yap, kcU and the like. IV, Omissions by Theon. Heiberg remarks that, Theon's object having been, as above shown, to amplify and explain Euclid, we should not naturally have expected to find him doing much in the contrary process of com- pression, and it is only owing to the recurrence of a certain sort of omissions so frequently (especially in the first Books) as to exclude the hypothesis of their being all due to chance that we are bound to credit him, with alterations making for greater brevity. We have seen, it is true, that he made omissions as well as additions for the purpose of reducing the language to a certain standard form. But there are also a good number of cases where in the enunciation of propositions, and in the exposition (the re-statement of them with reference to the figure), he has left out words because, apparently, he regarded Euclid's language as being too careful and precise. ch. v] THE TEXT 57 Again, he is apparently responsible for the frequent omission of the words oirtp ekei Setfot (or -iratijo-ai), Q.E.D. (or F.>. at the end of propositions. This is often the case at the end of porisms, where, in omitting the words, Theon seems to have deliberately departed from Euclid's practice. The MS. P seems to show clearly that, where Euclid put a porism at the end of a proposition, he omitted the Q.E.D. at the end of the proposition but inserted it at the end of the porism, as if he regarded the latter as being actually a part of the proposition itself. As in the Theonine MSS. the Q.ED, is generally omitted, the omission would seem to have been due to Theon. Sometimes in these cases the Q.E.D. is interpolated at the end of the proposition. Heiberg summed up the discussion of Theon 's edition by the remark that Theon evidently took no pains to discover and restore from MSS. the actual words which Euclid had written, but aimed much more at removing difficulties that might be feit by learners in studying the book. His edition is therefore not to be compared with the editions of the Alexandrine grammarians, but rather with the work done by Eutocius in editing 1 Apollonius and with an interpolated recension of some of the works of Archimedes by a certain Byzantine, Theon occupying a position midway between these two editors, being superior to the latter in mathematical knowledge but behind Eutocius in industry (these views now require to be some- what modified, as above stated). But however little Theon's object may be approved by those of us who would rather know the ipsissima verba of Euclid, there is no doubt that his work was approved by his pupils at Alexandria for whom it was written ; and his edition was almost exclusively used by later Greeks, with the result that the more ancient text is only preserved to us in one MS. As the result of the above investigation, we may feel satisfied that, where P and the Theonine MSS. agree, they give us (except in a few accidental instances) Euclid as he was read by the Greeks of the 4th c. But even at that time the text had been passed from hand to hand through more than six centuries, so that it is certain that it had already suffered changes, due partly to the fault of copyists and partly to the interpolations of mathematicians. Some errors of copyists escaped Theon and were corrected in some MSS. by later hands. Others appear in all our MSS. and, as they cannot have arisen accidentally in all, we must put them down to a common source more ancient than Theon. A somewhat serious instance is to be found in III. % ; and the use of aTrrka&w for e<j>aTrr4<r0<i> in the sense of " touch " may also be mentioned, the proper distinction between the words having been ignored as it was by Theon also. But there are a number of imperfections in the ante-Theonine text which it would be unsafe to put down to the errors of copyists, those namely where the good MSS. agree and it is not possible to see any motive that a copyist could have had for altering a correct reading. In these cases it is possible that the imperfections are due to a certain degree of carelessness on the part of Euclid himself; for it 58 INTRODUCTION [ch. v is not possible " Euclidem ab omni naevo vindicare," to use the words of Saccheri", and consequently Slmson is not right in attributing to Theon and other editors all the things in Euclid to which mathe- matical objection can be taken. Thus, when Euclid speaks of " the ratio compounded of the sides" for "the ratio compounded of the ratios of the sides," there is no reason for doubting that Euclid himself is responsible for the more slip-shod expression. Again, in the Books XI. — Xin. relating to solid geometry there are blots neither few nor altogether unimportant which can only be attributed to Euclid himself; and there is the less reason for hesitation in so attributing them because solid geometry was then being treated in a thoroughly systematic manner for the first time. Sometimes the conclusion (ovfvn-ipaa-pa) of a proposition does not correspond exactly to the enunciation, often it is cut short with the words teal t« i^rjf "and the rest" (especially From Book X. onwards), and very often in Books viil., IX. it is omitted. Where all the MSS. agree, there is no ground for hesitating to attribute the abbreviation or omission to Euclid; though, of course, where one or more mss. have the longer form, it must be retained because this is one of the cases where a copyist has a temptation to abbreviate. Where the true reading is preserved in one of the Theonine MSS. alone, Heiberg attributes the wrong reading to a mistake which arose before Theon's time, and the right reading of the single MS. to a successful correction. We now come to the most important question of the Interpolations introduced before Theon's time. I. Alternative proofs or additional cases. It is not in itself probable that Euclid would have given two proofs of the same proposition ; and the doubt as to the genuineness of the alternatives is increased when we consider the character of some of them and the way in which they are introduced. First of all, we have those of VI. 20 and XII. 1 7 introduced by " we shall prove this otherwise more readily (irpox^pOTepov)" or that of X. 90 " it is possible to prove more shortly (trvvTopcoTepov)." Now it is impossible to suppose that Euclid would have given one proof as that definitely accepted by him and then added another with the express comment that the latter has certain advantages over the former. Had he con- sidered the two proofs and come to this conclusion, he would have inserted the latter in the received text instead of the former. These alternative proofs must therefore have been interpolated. The same argument applies to alternatives introduced with the words "or even thiis " (tj Kal ovtbw), " or even otherwise " (1} teal a\ka><i). Under this head come the alternatives for the last portions of in. 7, 8 ; and Heiberg also compares the alternatives for parts of HI. 3 1 (that the angle in a semicircle is a right angle) and xill. 1 3, and the alternative proof of the lemma after X. 32. The alternatives to X. I OS and 106, 1 Kudidis ab &mm veuve vittdfcafus, Mediolani, 1733. * Cf, especially the assumption, without proof or definition, of the criterion for ifutU solid angles, and the incomplete proof of xtt. 17. ch. v] THE TEXT $9 again, are condemned by the place in which they occur, namely after an alternative proof to x. in;. The above alternatives being all admitted to be spurious, suspicion must necessarily attach to the few others which are in themselves unobjectionable, Heiberg instances the alternative proofs to 111.9,111. 10, VI. 30, vi. 31 and XI. 22, observing that it is quite comprehensible that any of these might have occurred to a teacher or editor and seemed to him, rightly or wrongly, to be better than the corresponding proofs in Euclid. Curiously enough, Simson adopted the alternatives to III. 9, 10 in preference to the genuine proofs. Since Heiberg's preface was written, his suspicion has been amply confirmed as regards tit. 10 by the commentary of an-Nairlsl (ed. Curtze) which shows not only that this alternative is Heron's, but also that the substantive proposition III. 12 in Euclid is also Heron's, having been given by him to supplement HI. 1 1 which must originally have been enunciated of circles "touching one another " simply, i.e. so as to include the case of external as well as internal contact, though the proof covered the case of internal contact only. "Euclid, in the nth proposition," says Heron, " supposed two circles touching one another internally and wrote the proposition on this case, proving what it was required to prove in it, But I will show how it is to be -proved if the contact be external 1 ." This additional proposition of Heron's is by way of adding another case, which brings us to that class of interpolation. It was the practice of Euclid and the ancients to give only one case (generally the most difficult one) and to leave the others to be investigated by the reader for himself. One interpolation of a second case (VI. 27) is due, as we have seen, to Theon, The two extra cases of XI. 23 were manifestly interpolated before Theon 's time, for the preliminary distinction of three cases, " (the centre) will either be within the triangle LMN, or on one of the sides, or outside. First let it be within," is a spurious addition (B and V only). Similarly an unnecessary case is interpolated in III. 11. II. Lemmas, Heiberg has unhesitatingly placed in his Appendix to Vol. HI. certain lemmas interpolated either by Theon (on X. 13) or later writers (on X. 27, 29, 31, 32, 33, 34, where V only has the lemmas). But we are here concerned with the lemmas found in all the MSS., which however are, for different reasons, necessarily suspected. We will deal with the Book x. lemmas last. (1) There is an a priori ground of objection to those lemmas which come after the propositions to which they relate and prove properties used in those propositions ; for, if genuine, they would be a sign of faulty arrangement such as would not be likely in a systematic work so carefully ordered as the Elements. The lemma to VI. 22 is one of this class, and there is the further objection to it that in VI. 28 Euclid makes an assumption which would equally require a lemma though none is found. The lemma after XII. 4 is open to the further objections that certain altitudes are used but are not drawn in the 1 An-NairUl, ed. Curtze, p. ill. 6o INTRODUCTION {ch. v figure (which is not in the manner of Euclid), and that a peculiar expression " parallelepi pedal solids described on {avaypafyofiGva airo) prisms " betrays a hand other than Euclid's. There is an objection on the score of language to the lemma aftc XIII. 2, The lemmas on xi. 23, XIII. 13, XIII. 18, besides coming after the propositions to which they relate, are not very necessary in themselves and, as regards the lemma to XIII. 13, it is to be noticed that the writer of a gloss in the proposition could not have had it, and the words "as will be proved afterwards " in the text are rightly suspected owing to differences between the MS. readings. The lemma to XII. 2 also, to which Simson raised objection, comes after the proposition ; but, if it is rejected, the words " as was proved before " used in XII, 5 and 1 8, and referring to this lemma, must be struck out. (2) Reasons of substance are fatal to the lemma before X. 60, which is really assumed in X. 44 and therefore should have appeared there if anywhere, and to the lemma on X. 20, which tries to prove what is already stated in X. Def. 4. We now come to the remaining lemmas in Book X., eleven in number, which come be/ore the propositions to which they relate and remove difficulties in the way of their demonstration. That before X. 42 introduces a set of propositions with the words " that the said irrational straight lines are uniquely divided... we will prove after premising the following lemma," and it is not possible to suppose that these words are due to an interpolator ; nor are there any objections to the lemmas before X. 14, 17, 22, 33, 54, except perhaps that they are rather easy. The lemma before X. 10 and X. 10 itself should probably be removed from the Elements ; for X. 10 really uses the following proposition X. 1 1, which is moreover numbered 10 by the firsthand in P, and the words in x. 10 referring to the lemma "for we learnt (how to do this)" betray the interpolator. Heiberg gives reason also for rejecting the lemmas before x. 19 and 24 with the words " in any of the aforesaid ways " (omitted in the Theonine MSs.) in the enunciations of X. 19, 24 and in the exposition of X. 20. Lastly, the lemmas before x. 29 may be genuine, though there is an addition to the second of them which is spurious. Heiberg includes under this heading of interpolated lemmas two which purport to be substantive propositions, XI. 38 and XIII. 6. These must be rejected as spurious for reasons which will be found in detail in my notes on XL 37 and XIII. 6 respectively. The latter proposition is only quoted once (in xin. 17) ; probably the words quoting it (with ypa/ifty instead of ti/ffeta) are themselves interpolated, and Euclid thought the fact stated a sufficiently obvious inference from xin. 1. III. Porisms (or corollaries). Most of the porisms in the text are both genuine and necessary ; but some are shown by differences- in the MSS. not to be so, e.g. those to 1. 15 (though Prod us has it), in. 31 and vi. 20 (For. 2). Sometimes parts of porisms are interpolated. Such are the last few lines in the porisms to iv. 5, vi. 8 ; the latter addition is proved later by ch.v] THE TEXT 61 means of VI. 4, 8, so that the writer of these proofs could not have had the addition to vi. 8 Por. before him. Lastly, interpolators have added a sort of proof to some porisms, as though they were not quite obvious enough ; but to add a demonstration is inconsistent with the idea of a porism, which, according to Proclus, is a by-product of a proposition appearing without our seeking it IV. Scholia. Several interpolated scholia betray themselves by their wording, e.g. those given by Heiberg in the Appendix to Book X. and contain- ing the words icaket, ixaXeve ("he calls" or "called"); these scholia were apparently written as marginal notes before Theon's time, and, being adopted as such by Theon, found their way into the text in P and some of the Theon ine MSS. The same thing no doubt accounts for the interpolated analyses and syntheses to xm. 1 — 5, as to which see my note on xin. 1. V. Interpolations in Book X. First comes the proposition " Let it be proposed to us to show that in square figures the diameter is incommensurable in length with the side, ' which, with a scholium after it, ends the tenth Book. The form of the enunciation is suspicious enough and the proposition, the proof of which is indicated by Aristotle and perhaps was Pythagorean, is perfectly unnecessary when X, 9 has preceded. The scholium ends with remarks about commensurable and incommensurable solids, which are of course out of place before the Books on solids. The scholiast on Book x. alludes to this particular scholium as being due to " Theon and some others." But it is doubtless much more ancient, and may, as Heiberg conjectures have been the beginning of Apollonius' more advanced treatise on incom men su rabies. Not only is everything in Book X. after x. 1 1 5 interpolated, but Heiberg doubts the genuineness even of X. 112 — 115, on the ground that X, m rounds off the theory of incommensu rabies as we want it in the Books on solid geometry, while X 112 — 115 are not really connected with what precedes, nor wanted for the later Books, but seem to form the starting-point of a new and more elaborate theory of irrationals, VI. Other minor interpolations are found of the same character as those above attributed to Theon. First there are two places (xi, 35 and XI. 26) where, after "similarly we shail prove" and "for the same reason," an actual proof is nevertheless given. Clearly the proofs are interpolated ; and there are other similar interpolations. There are also interpolations of intermediate steps in proofs, unnecessary explanations and so on, as to which I need not enter into details. Lastly, following Heiberg's order, I come to VII. Interpolated definitions, axioms etc. Apart from VI. Def. 5 (which may have been interpolated by Theon although it is found written in the margin of P by the first hand), the definition of a segment of a circle in Book 1. is interpolated, as is clear from the fact that it occurs in a more appropriate place in Book til. and Proclus omits it. VI. Def. 2 (reciprocal figures) is rightly condemned by Simson — perhaps it was taken from Heron — and 6* INTRODUCTION [ch. v Heiberg would reject vii. Def, 10, as to which see my note on that definition. Lastly the double definition of a solid angle (XL Def. n) constitutes a difficulty. The use of the word iirifydveia suggests that the first definition may have been older than Euclid, and he may have quoted it from older elements, especially as his own definition which follows only includes solid angles contained by planes, whereas the other includes other sorts (cf. the words ypaftpSv, ypaftpaZ<;) which are also distinguished by Heron (Def. 22). If the first definition had come last, it could have been rejected without hesitation : but it is not so easy to reject the first part up to and including " otherwise " (aXAw?). No difficulty need be felt about the definitions of "oblong," "rhombus," and "rhomboid," which are not actually used in the Elements; they were no doubt taken from earlier elements and given for the sake of completeness. As regards the axioms or, as they are called in the text, common notions (Kotval evvotai), it is to be observed that Proclus says 1 that Apollonius tried to prove "the axioms," and he gives Apollonius' attempt to prove Axiom i. This shows at all events that Apollonius had some of the axioms now appearing in the text. But how could Apollonius have taken a controversial line against Euclid on the subject of axioms if these axioms had not been Euclid's to his know- ledge? And, if they had been interpolated between Euclid's time and his own, how could Apollonius, living so comparatively short a time after Euclid, have been ignorant of the fact ? Therefore some of the axioms are Euclid's (whether he called them common notions, or axioms, as is perhaps more likely since Proclus calls them axioms) : and we need not hesitate to accept as genuine the first three discussed by Proclus, viz. (i) things equal to the same equal to one another, (2) if equals be added to equals, wholes equal, (3) if equals be subtracted from equals, remainders equal. The other two mentioned by Proclus (whole greater than part, and congruent figures equal) are more doubtful, since they are omitted by Heron, Martianus Capella, and others. The axiom that "two lines cannot enclose a space" is however clearly an interpolation due to the fact that I. 4 appeared to require it. The others about equals added to unequals, doubles of the same thing, and halves of the same thing are also interpolated ; they are connected with other interpolations, and Proclus clearly used some source which did not contain them. Euclid evidently limited his formal axioms to those, which seemed to him most essential and of the widest application ; for he not un- frequently assumes other things as axiomatic, e.g. in vii. 28 that, tf a number measures two numbers, it measures their difference. The differences of reading appearing in Proclus suggest the question of the comparative purity of the sources used by Proclus, Heron and others, and of our text. The omission of the definition of a segment in Book I. and of the old gloss "which is called the cir- cumference" in I. Def. 15 (also omitted by Heron, Taurus, Sextus 1 Proclus, pp. 194, losqq. ch. v] THE TEXT 63 Empiricus and others) indicates that Prod us had better sources than we have ; and Heiberg gives other cases where Proclus omits words which are in all our MSS. and where Proclus' reading should perhaps be preferred. But, except in these instances (where Proclus may have drawn from some ancient source such as one of the older com- mentaries), Proclus' MS. does not seem to have been among the best. Often it agrees with our worst mss., sometimes it agrees with F where F alone has a certain reading in the text, so that (e.g. in I. 15 Por.) the common reading of Proclus and F must be rejected, thrice only does it agree with P alone, sometimes it agrees with P and some Theonine mss., and once it agrees with the Theonine mss. against P and other sources. Of the other external sources, those which are older than Theon generally agree with our best MSS., e.g. Heron, allowing for the difference in the plan of his definitions and the somewhat free adap- tation to his purpose of the Euclidean definitions in Books X., XI. Heiberg concludes that the Elements were most spoiled by inter- polations about the 3rd c, for Sextus Empiricus had a correct text, while Iamblichus had an interpolated one; but doubtless the purer text continued for a long time in circulation, as we conclude from the fact that our MSS. are free from interpolations already found in Iamblichus' MS. CHAPTER VI. THE SCHOLIA. Heiberg has collected scholia, to the number of about 1500, in Vol. V. of his edition of Euclid, and has also discussed and classified them in a separate short treatise, in which he added a few others 1 . These scholia cannot be regarded as doing much to facilitate the reading of the Elements. As a rule, they contain only such observa- tions as any intelligent reader could make for himself. Among the few exceptions are XI. Nos. 33, 35 (where XI. 22, 23 are extended to solid angles formed by any number of plane angles), xil. No. 85 (where an assumption tacitly made by Euclid in XII. 17 is proved), IX. Nos. 28, 29 (where the scholiast has pointed out the error in the text of IX. 19). Nor are they very rich in historical information ; they cannot be compared in this respect with Proclus' commentary on Book I. or with those of Eutocius on Archimedes and Apollonius. But even under this head they contain some things of interest, e.g. II. No. 1 1 explaining that the gnomon was invented by geometers for the sake of brevity, and that its name was suggested by an incidental characteristic, namely that "from it the whole is known (yvapl^Tai), either of the whole area or of the remainder, when it (the yva>ftmv) is either placed round or taken away"; 11. No, 13, also on the gnomon; IV. No. 2 stating that Book IV. was the discovery of the Pythagoreans ; V. No. 1 attributing the content of Book v. to Eudoxus ; x. No. 1 with its allusion to the discovery of incommensurability by the Pytha- goreans and to Apollonius' work on irrationals; x. No. 62 definitely attributing X. 9 to Theaetetus; XIII. No. I about the "Platonic" figures, which attributes the cube, the pyramid, and the dodecahedron to the Pythagoreans, and the octahedron and icosahedron to Theaetetus. Sometimes the scholia are useful in connexion with the settlement of the text, (1) directly, e.g. III. No. 16 on the interpolation of the word "within" (eWot) in the enunciation of 111, 6, and x. No. 1 alluding to the discussion by "Theon and some others" of irrational "surfaces" and "solids," as well as "lines," from which we may 1 Heiberg, Om Schelitrne til Eutlids Elemenitr, Kjebenhavn, 1888. The tract is written in Danish, but, fortunately for those who do not read Danish easily, the author has appended (pp. 70 — 78) a resume in French. ch. vi] THE SCHOLIA 65 conclude that the scholium at the end of Book x, is not genuine ; (2) indirectly in that they sometimes throw light on the connexion of certain MSS. Lastly, they have their historical importance as enabling us to judge of the state of mathematical science at the times when they were written. Before passing to the classification of the scholia, Heiberg remarks that we must separate from them a number of additions in the nature of scholia which are found in the text of our MSS. but which can, in one way or another, be proved to be spurious. As they are found both in P and in the Theonine MSS., they must have been in the MSS. anterior to Theon (4th c). But they are, in great part, only found in the margin of P and the Theonine MSS. ; in V they are half in the text and half in the margin. This can hardly be explained except on the supposition that these additions were originally (in the MSS. before Theory) in the margin, and that Theon kept them there in his edition, but that they afterwards found their way gradually into the text of P as well as of the Theonine MSS., or were omitted altogether, while particular MSS. have in certain places preserved the old arrange- ment Of such spurious additions Heiberg enumerates the following: the axiom about equals subtracted from unequals, the last lines of the porism to vi. 8, second porisms to v. 19 and to vi. 20, the porism to hi. 31, vi. Def. 5, various additions in Book X., the analyses and syntheses of XI 1 1. 1 — Si an ^ the proposition XIII. 6. The two first classes of scholia distinguished by Heiberg are denoted by the convenient abbreviations "Schol. Vat." and "Schol. Vind." I. Schol, Vat. It is first necessary^ to set out the letters by which Heiberg denotes certain collections of scholia. P = Scholia in P written by the first hand. B = Scholia in B by a hand of the same date as the MS. itself, generally that of Aretha s. F = Scholia in F by the first hand. Vat = Scholia of the Vatican MS. 204 of the 10th c, which has these scholia on leaves 198 — 205 (the end is missing) as an independent collection. It does not contain the text of the Elements. V c = Scholia found on leaves 283 — 292 of V and written in the same hand as that part of the MS. itself which begins at leaf 233. Vat 192 = a Vatican MS. of the 14th c. which contains, after (l) the Elements I. — XIII. (without scholia), (2) the Data with scholia, (3) Marinus on the Data, the Schol. Vat as an independent collection and in their entirety, beginning with 1. No. 88 and ending with xm. No. 44. The Schol. Vat., the most ancient and important collection of scholia, comprise those which are found in PBF Vat. and, from VII, 12 to IX. 15, in PB Vat. only, since in that portion of the Elements F was restored by a later hand without scholia; they also include 1, 66 INTRODUCTION [ch. vi No. 88 which only happens to be erased in F, and IX. Nos. 28, 29 which may be left out because F, here has a different text In F and Vat. the collection ends with Book x. ; but it must also include Schol. FB of Books xi. — xill., since these are found along with Schol. Vat. to Books I. — X. in several MSS. (of which Vat. 192 is one) as a separate collection. The Schol. Vat. to Books X. — XIII. are also found in the collection V c (where, curiously enough, xill. Nos, 43, 44 are at the beginning). The Schol. Vat. accordingly include Schol. PBV C Vat, 192, and doubtless also those which are found in two of these sources. The total number of scholia classified by Heiberg as Schol. Vat. is 138. As regards the contents of Schol. Vat. Heiberg has the following observations. The thirteen scholia to Book I. are extracts made from Proclus by a writer thoroughly conversant with the subject, and cleverly recast (with some additions). Their author does not seem to have had the two lacunae which our text of Proclus has (at the end of the note on 1. 36 and the beginning of the next note, and at the beginning of the note on I. 43), for the scholia I. Nos. 125 and 137 seem to fill the gaps appropriately, at least in part. In some passages he had better readings than our MSS. have. The rest of Schol. Vat. (on Books II. — xill.) are essentially of the same character as those on Book 1., containing prolegomena, remarks on the object of the propositions, critical remarks on the text, converses, lemmas ; they are, in general, exact and true to tradition. The reason of the resemblance between them and Proclus appears to be due to the fact that they have their origin in the commentary of Pappus, of which we know that Proclus also made use. In support of the view that Pappus is the source, Heiberg places some of the Schol, Vat. to Book X. side by side with passages from the com- mentary of Pappus in the Arabic translation discovered by Woepcke 1 ; he also refers to the striking confirmation afforded by the fact that XII, No. 2 contains the solution of the problem of inscribing in a given circle a polygon similar to a polygon inscribed in another circle, which problem Eutocius says' that Pappus gave in his commentary on the Elements. But, on the other hand, Schol. Vat. contain some things which cannot have come from Pappus, e.g. the allusion in X. No. 1 to Theon and irrational surfaces and solids, Theon being later than Pappus ; in. No. 10 about porisms is more like Proclus' treatment of the subject than Pappus', though one expression recalls that of Pappus about forming (c^Tj/MtTifeo-ftit) the enunciations of porisms like those of either theorems or problems. The Schol, Vat. give us important indications as regards the text of the EUmtntt as Pappus had it. In particular, they show that he could not have had in his text certain of the lemmas in Book X. For example, three of these are identical with what we find in Schol, 1 Om Scholitnu til Euklids EUmtnltr, pp. 11, it: cf. Euilid-StHtiien, pp. 170, 171; Woepcke, MJmoira prtstnt. & ?Ac«d. dts Seitnnt, 18 56, XIV. p. 6j8sqq. * Archimedes, ed. Heiberg, in. p. a8, 19—15. CM. VI] THE SCHOLIA 67 Vat (the lemma to X. 17 = Schol. X. No. 106, and the lemmas to X. 54, 60 come in Schol. X. No. 328) ; and it is not possible to suppose. that these lemmas, if they were already in the text, would also be given as scholia. Of these three lemmas, that before X. 60 has already been condemned for other reasons ; the other two, un- objectionable in themselves, must be rejected on the ground now stated. There were four others against which Heiberg found nothing to urge when writing his prolegomena to Vol. v., viz. the lemmas before X. 42, X. 14, X. 2Z and X. 33. Of these, the lemma to X. 22 is not reconcilable with Schol, x. No. 161, which takes up the assumption in the text of Eucl. X. 22 as if no lemma had gone before. The lemma to X. 42, which, on account of the words introducing it (see p. 60 above), Heiberg at first hesitated to regard as an inter- polation, is identical with Schol. X. No. 27a It is true that in Schol. x. No. 269 we find the words "this lemma has been proved before (£p tow ipTrpoaStv), but it shall also be proved now for convenience' sake (rov eroinov evexa,)" and it is possible to suppose that " before " may mean in Euclid's text before x. 42 ; but a proof in that place would surely have been as " convenient " as could be desired, and it is therefore more probable that the proof had been given by Pappus in some earlier place. (It may be added that the lemma to X. 14, which is identical with the lemma to XI. 23, con- demned on other grounds, is for that reason open to suspicion.) Heiberg's conclusion is that all the lemmas are spurious, and that most or alt of them have found their way into the text from Pappus' commentary, though at a time anterior to Theon's edition, since they are found in all our MSS. This enables us to fix a date for these interpolations, namely the first half of the 4th c. Of course Pappus had not in his text the interpolations which, from the fact of their appearing only in some of our MSS., are seen to be later than those above-mentioned. Such are the lemmas which are found in the text of V only after X. 29 and X. 31 respectively and are given in Heiberg's Appendix to Book X. (numbered 10 and 11). On the other hand it appears from Woepcke's tract 1 that Pappus already had x, 115 in his text : though it does not follow from this that the proposition is genuine but only that interpolations began very early. Theon interpolated a proposition (or lemma) between X. 12 and X. 13 (No, S in Heiberg's Appendix). Schol. Vat. has the same thing (X. No. 125). The writer of the scholia therefore did not find this lemma in the text. Schol. Vat IX. Nos. 28, 29 show that neither did he find in his text the alterations which Theon made in Eucl. IX. 19; the scholia in fact only agree with the text of P, not with Theon's. This suggests that Schol. Vat. were written for use with a MS. of the ante-Theonine recension such as F is. This probability is further confirmed by a certain independence which P shows in several places when compared with the Theon ine MSS. Not only has P better readings in some passages, but more substantial divergences; and, 1 Woepcke, op, rft. p. 702. 68 INTRODUCTION [eft vi in particular, the absence in P of three notes of a historical character which are added, wholly or partly from Prod us, in the Theonine MSS. attests an independent and more primitive point of view in P. In view of the distinctive character of P, it is possible that some of the scholia found in it in the first hand, but not in the other sources of Schol. Vat., also belong to that collection ; and several circumstances confirm this. Schol. XIII. No. 45, found in P only, which relates to a passage in Eucl. XIII. 13, shows that certain words in the text, though older than Theon, are interpolated ; and, as the scholium is itself older than Theon, is headed "third lemma," and follows a "second lemma" relating to a passage in the text im- mediately preceding, which "second lemma" belongs to Schol. Vat. and is taken from Pappus, the "third" in all probability came from Pappus also. The same is true of Schol. XII. No. 72 and xm. No. 69, which are respectively identical with the propositions vulgo XI. 38 (Heiberg, A pp. to Book xi., No, 3) and XII I. 6; for both of these interpolations are older than Theon. Moreover most of the scholia which P in the first hand alone has are of the same character as Schol, Vat Thus VII. No. 7 and XIII. No. I introducing Books VII. and xm. respectively are of the same historical character as several of Schol Vat ; that vil. No. 7 appears in the text of P at the beginning of Book VII. constitutes no difficulty. There are a number of converses, remarks on the relation of propositions to one another, explanations such as XII. No. 89 in which it is remarked that <f>, fl in Euclid's figure to xil. 17 {Z, V in my figure) are really the same point but that this makes no difference in the proof. Two other Schol. P on XII. 17 are connected by their headings with XII. No. 72 mentioned above, xi. No. 10 (P) is only another form of xi. No. 1 1 (B) ; and B often, alone with P, has preserved Schol. Vat On the whole Heiberg considers some 40 scholia found in P alone to belong to Schol. Vat. The history of Schol. Vat. appears to have been, in its main outlines, the following. They were put together after 500 A.D., since they contain extracts from Proclus, to which we ought not to assign a date too near to that of Proclus' work itself; and they must at least be earlier than the latter half of the 9th c, in which B was written. As there must evidently have been several intermediate links between the archetype and B, we must assign them rather to the first half of the period between the two dates, and it is not improbable that they were a new product of the great development of mathematical studies at the end of the 6th c. (Isidorus of Miletus). The author extracted what he found of interest in the commentary of Proclus on Book I. and in that of Pappus on the rest of the work, and put these extracts in the margin of a MS. of the class of P. As there are no scholia to I. 1 — 22, the first leaves of the archetype or of one of the earliest copies must have been lost at an early date, and it was from that mutilated copy that partly P and partly a MS. of the Theonine class were taken, the scholia being put in the margin in both. Then the collection spread through the Theonine MSS., gradually losing some ch. vi] THE SCHOLIA 69 scholia which could not be read or understood, or which were accidentally or deliberately omitted. Next it was extracted from one of these MSS. and made into a separate work which has been preserved, in part, in its entirety (Vat. 192 etc.) and, in part, divided into sections, so that ihe scholia to Books X. — xni. were detached (V c ). It had the same fate in the mss, which kept the original arrangement (in the margin), and in consequence there are some MSS. where the scholia to the stereometric Books are missing, those Books having come to be less read in the period of decadence. It is from one of these MSS. that the collection was extracted as a separate work such as we find it in Vat. ( roth c). II. The second great division of the scholia is Schol. Vind. This title is taken from the Viennese MS. (V), and the letters used by Heiberg to indicate the sources here in question are as follows. V* = scholia in V written by the same hand that copied the MS. itself from fol. 235 onward. q = scholia of the Paris MS. 2344 (q) written by the first hand. 1 = scholia of the Florence ms. Laurent xxvin, 2 written in the 13th — 14th c, mostly in the first hand, but partly in two later hands. V b = scholia in V written by the same hand as the first part (leaves 1 — 1S3) of the MS. itself; V" wrote his scholia after V". q 1 = scholia of the Paris MS. (q) found here and there in another hand of early date. Schol. Vind. include scholia found in V m q. 1 is nearly related to q ; and in fact the three Mss. which, so far as Euclid's text is con- cerned, show no direct interdependence, are. as regards their scholia, derived from one original. Heiberg proves this by reference to the readings of the three in two passages (found in Schol. I, No. 109 and X. No, 39 respectively). The common source must have contained, besides the scholia found in the three MSS. V a ql, those also which are contained in two of them, for it is more unlikely that two of the three should contain common interpolations than that a particular scholium should drop out of one of them. Besides V" and q, the scholia V b and q 1 must equally be referred to Schol. Vind., since the greater part of their scholia are found in 1. There is a lacuna in q from Eucl. VIII. 25 to IX. 14, so that for this portion of the Elements Schol. Vind. are represented by VI only, Heiberg gives about 450 numbers in all as belonging to this collection. Schol. Vind. did not all come from one source; this is shown by differences of substance, e.g. between X. Nos. 36 and 39, and by differences of time of writing : e.g. vi. No. 52 refers at the beginning to No. 55 with the words "as the scholium has it" and is therefore later than that scholium ; X. No. 247 is also later than x. No. 246. The scholia to Book I. are here also extracts from Proclus, but more copious and more verbatim than in Schol. Vat. The author has not always understood Proclus; and he had a text as bad as that of our MSS., with the same lacunae. The scholia to the other 7 o INTRODUCTION [ch. VI Books are partly drawn (i) from Schol. Vat., the MSS. representing Schol. Vind, and Schol. Vat. in these cases showing nearly all possible combinations; but there is no certain trace in Schol. Vind. of the scholia peculiar to P. The author used a copy of Schol. Vat. in the form in which they were attached to the Theonine text ; thus Schol. Vind. correspond to BF Vat., where these diverge from P, and especially closely to B. Besides Schol. Vat., the editors of Schol. Vind. used {2) other old collections 0/ scholia of which we find traces in B and F; Schol. Vind. have also some scholia common with b. The scholia which Schoi. Vind. have in common with BF come from two different sources, and were apparently afterwards introduced into the other MSS. ; one result of this is that several scholia are reproduced twice. But, besides the scholia derived from these sources, Schol. Vind. contain a large number of others of late date, characterised by in- correct language or by triviality of content (there are many examples in numbers, citations of propositions used, absurd diroplai, and the like). Unlike Schol. Vat, these scholia often quote words from Euclid as a heading (in one case a heading is inserted in Schol. Vind. where a scholium without the heading is quoted from Schol. Vat, see V. No. 14). The explanations given often presuppose very little know- ledge on the part of the reader and frequently contain obscurities and gross errors. Schol. Vind. were collected for use with a MS. of the Theonine class; this follows from the fact that they contain a note on the proposition vulgo VII. 22 interpolated by Theon (given in Heiberg's App. to Vol. II. p. 430), Since the scholium to vn. 39 given in V and p in the text after the title of Book VIII. quotes the proposition as VII. 39, it follows that this scholium must have been written before the interpolation of the two propositions vulgo VII. 20, 22 ; Schol. Vind. contain (vn. No. 80) the first sentence of it, but without the heading referring to VII. 39. Schol. VII. No. 97 quotes VII. 33 as VII. 34, so that the proposition vulgo vn. 22 may have stood in the scholiast's text but not the later interpolation vulgo vn. 20 (later because only found in B in the margin by the first hand). Of course the scholiast had also the interpolations earlier than Theon. For the date of the collection we have a lower limit in the date (12th c.) of MSS. in which the scholia appear. That it was not much earlier than the 12th c. is indicated (1) by the poverty of its contents, (2) by the quality of the ms. of Proclus which was used in the compilation of it (the Munich MS. used by Friedlein with which the scholiast's excerpts are essentially in agreement belongs to the I ith — 12th c), (3) by the fact that Schol. Vind. appear only in MSS. of the 12th c. and no trace of them is found in our MSS. belonging to the 9th — 10th c. in which Schol. Vat. are found. The collection may therefore probably be assigned to the 1 ith c. Perhaps it may be in part due to Psellus who lived towards the end of that century : for in a Florence MS. (Magliabecch. XI, 53 of the 15 th c.) containing a mathematical compendium intended for use in the reading of Aristotle ch. vi] THE SCHOLIA 71 the scholia i. Nos. 40 and 49 appear with the name of Psellus attached. Schoi. Vind. are not found without the admixture of foreign elements in any of our three sources. In 1 there are only very few such in the first hand. In q there are several new scholia in the first hand, for the most part due to the copyist himself. The collection of scholia on Book x. in q (Heiberg's q=) is also in the first hand ; it is not original, and it may perhaps be due to Psellus (Maglb. has some definitions of Book x. with a heading "scholia of... Michael Psellus on the definitions of Euclid's 10th Element" and Schol. X. No. 9), whose name must have been attached to it in the common source of Maglb. and q ; to a great extent it consists of extracts from Schol. Vind. taken from the same source as VI. The scholia q 1 (in an ancient hand in q), confined to Book II., partly belong to Schol, Vind. and partly correspond to b 1 (Bologna MS.), q* and q b are in one hand (Theodorus Antiochita), the nearest to the first hand of q ; they are doubtless due to an early possessor of the MS. of whom we know nothing more. V* has, besides Schol. Vind., a number of scholia which also appear in other MSS., one in BFb, some others in P, and some in v (Codex Vat. IO38, 13th c.) ; these scholia were taken from a source in which many abbreviations were used, as they were often misunderstood by V 1 . Other scholia in V" which are not found in the older sources — some appearing in V* alone— are also not original, as is proved by mistakes or corruptions which they contain ; some others may be due to the copyist himself. V b seldom has scholia common with the other older sources ; for the most part they either appear in V b alone or only in the later sources as v or F* (later scholia in F), some being original, others not. In Book X. V b has three series of numerical examples, ( 1 ) with Greek numerals, (2) alternatives added later, also mostly with Greek numerals, (3) with Arabic numerals. The last class were probably the work of the copyist himself. These examples (cf. p. 74 below) show the facility with which the Byzantines made calculations at the date of the MS. (12th c). They prove also that the use of the Arabic numerals (in the East- Arabian form) was thoroughly established in the 1 2th c. ; they were actually known to the Byzantines a century earlier, since they appear, in the first hand, in an Escurial MS. of the 1 1 th c. Of collections in other hands in V distinguished by Heiberg (see preface to Vol. v.), V 1 has very few scholia which are found in other sources, the greater part being original ; V ! , V s are the work of the copyist himself; V* are so in part only, and contain several scholia from Schol. Vat. and other sources. V* and V J are later than 13th — 14th c, since they are not found in f (cod. Laurent XXVHl, 6) which was copied from V and contains, besides V" V b , the greater part of V 1 and vi. No. 20 of V (in the text). In P there are, besides P* (a quite late hand, probably one of the old Scriptores Grace i at the Vatican), two late hands (P 1 ), one of which has some new and independent scholia, while the other has 7* INTRODUCTION [ch. vi added the greater part of Schol. Vind., partly in the margin and partly on pieces of leaves stitched on. Our sources for Schol. Vat. also contain other elements. In P there were introduced a certain number of extracts from Proclus, to supplement Schol. Vat. to Book I. ; they are all written with a different ink from that used for the oldest part of the MS., and the text is inferior. There are additions in the other sources of Schol. Vat. (F and B) which point to a common source for FB and which are nearly all found in other mss., and, in particular, in Schol. Vind,, which also used the same source ; that they are not assignable to Schol. Vat. results only from their not being found in Vat. Of other additions in F, some are peculiar to F and some common to it and b; but they are not original. F s (scholia in a later hand in F) contains three original scholia ; the rest come from V. B contains, besides scholia common to it and F, b or other sources, several scholia which seem to have been put together by Arethas, who wrote at least a part of them with his own hand. Heiberg has satis6ed himself, by a closer study of b, that the scholia which he denotes by b, ji and b 1 are by one hand ; they are mostly to be found in other sources as well, though some are original. By the same hand (Theodoras Cabasilas, 15th c.) are also the scholia denoted by b", B', b* and B ! . These scholia come in great part from Schol. Vind., and in making these extracts Theodoras probably used one of our sources, 1, mistakes in which often correspond to those of Theodoras. To one scholium is attached the name of Demetrius (who must be Demetrius Cydonius, a friend of Nicolaus Cabasilas, 14th c); but it could not have been written by him, since it appears in B antl Schol. Vind. Nor are all the scholia which bear the name of Theodoras due to Theodoras himself, though some are so. As B' (a late hand in B) contains several of the original scholia of b*, B* must have used b itself as his source, and, as all the scholia in B* are in b, the latter is also the source of the scholia in B 8 which are found in other MSS. B and b were therefore, in the 15th c, in the hands of the same person ; this explains, too, the fact that b in a late hand has some scholia which can only come from B. We arrive then at the conclusion that Theodoras Cabasilas, in the 15th c, owned both the MSS. B and b, and that he transferred to B scholia which he had before written in b, either independently or after other sources, and inversely transferred some scholia from B to b. Further, B' are earlier than Theodoras Cabasilas, who certainly himself wrote B* as well as b' and b 8 . An author's name is also attached to the scholia VI. No. 6 and X. No. 223, which are attributed to Maximus Planudes (end of 13th c) along with scholia on I. 31, x. 14 and X. 18 found in 1 in a quite late hand and published on pp. 46, 47 of Heiberg's dissertation. These seem to have been taken from lectures of Planudes on the Elements by a pupil who used 1 as his copy. There are also in 1 two other Byzantine scholia, written by a late hand, and bearing the names Ioannes and Pediasimus respectively ; ch. vi] THE SCHOLIA 73 these must in like manner have been written by a pupil after lectures of Ioannes Pediasimus (first half of 14th c), and this pupil must also have used 1. Before these scholia were edited by Heiberg, very few of them had been published in the original Greek. The Basel editio princeps has a few (v. No. 1, VI. Nos. 3, 4 and some in Book X.) which are taken, some from the Paris MS. (Paris. Gr, 2343) used by Grynaeus, others probably from the Venice MS. (Marc. 301) also used by him; one published by Heiberg, not in his edition of Euclid but in his paper on the scholia, may also be from Venet. 301, but appears also in Paris. Gr. 2342. The scholia in the Basel edition passed into the Oxford edition in the text, and were also given by August in the Appendix to his Vol. II. Several specimens of the two series of scholia (Vat. and Vind.) were published by C. Wachsmuth {Rhein, Mus. xvm, p. 132 sqq.) and by Knoche {Untersuchungen iiber die neu aufgefundenen Scholien des Proklus, Herford, 1 865). The scholia published in Latin were much more numerous. G. Valla {De expetendis et fugiendis rebus, 1 501) reproduced apparently some 200 of the scholia included in Heiberg's edition. Several of these he obtained from two Modena MSS. which at one time were in his possession (Mutin. Ill B, 4 and II E, 9, both of the 15th c.) ; but he must have used another source as well, containing extracts from other series of scholia, notably Schol. Vind. with which he has some 87 scholia in common. He has also several that are new. Commandinus included in his translation under the title "Scholia antiqua " the greater part of the Schol, Vat. which he certainly obtained from a MS. of the class of Vat. 192; on the whole he adhered closely to the Greek text. Besides these scholia Com- mandinus has the scholia and lemmas which he found in the Basel editio princeps, and also three other scholia not belonging to Schol. Vat., as well as one new scholium (to Xii. 13) not included in Heiberg's edition, which are distinguished by different type and were doubtless taken from the Greek MS. used by him along with the Basel edition. In Conrad Dasypodius' Lexicon matkematicum published in 1573 there is (on fol. 42—44) "Graecum scholion in definitiones Euclidis libri quinti elementorum append ids loco propter pagellas vacantes annexum." This contains four scholia, and part of two others, published in Heiberg's edition, with some variations of readings, and with some new matter added (for which see pp. 64 — 6 of Heiberg's pamphlet). The source of these scholia is revealed to us by another work of Dasypodius, haaci Monachi Scholia in Euclidis elementorum geometriae sex prions tibros per C. Dasypodium in latinum sermonem trans lata et in lucem edita (1579). This work contains, besides excerpts from Proclus on Book I. (in part closely related to Schol. Vind.), some 30 scholia included in Heiberg's edition, several new scholia, and the above-mentioned scholia to the definitions of Book v. published in Greek in 1573. After the scholia follow " Isaaci Monachi 74 INTRODUCTION [ch. vi prolegomena in Euclidis Elementorum geometriae libros" (two definitions of geometry) and " Varia miscellanea ad geometriae cogni- tionem necessaria ab Isaaco Monacho collecta " (mostly the same as pp. 252, 24 — 272, 27 in the Varieu Collectiones included in Hultsch's Heron) ; lastly, a note of Dasypodius to the reader says that these scholia were taken "ex clarissimi viri Joannis Sambuciantiquocodice manu propria Isaaci Monachi scrip to." Isaak Monachus is doubtless Isaak Argyrus, 14th c. ; and Dasypodius used a MS. in which, besides the passage in Hultsch's Variae CoIUctiotus, there were a number of scholia marked in the margin with the name of Isaak (cf. those in b under the name of Theodorus Cabasilas). Whether the new scholia are original cannot be decided until they are published in Greek ; but it is not improbable that they are at all events independent arrange- ments of older scholia. All but five of the others, and all but one of the Greek scholia to Book V., are taken from Schol. Vat. ; three of the excepted ones are from Schol, Vind., and the other three seem to come from F (where some words of them are illegible, but can be supplied by means of Mut. Ill B, 4, which has chese three scholia and generally shows a certain likeness to Isaak's scholia). Dasypodius also published in 1564 the arithmetical commentary of Barlaam the monk (14th c.) on Eucl. Book 11., which finds a place in Appendix IV. to the Scholia in Heiberg's edition. Hultsch has some remarks on the origin of the scholia 1 . He observes that the scholia to Book I. contain a considerable portion of Geminus' commentary on the definitions and are specially valuable because they contain extracts from Geminus only, whereas Proclus, though drawing mainly upon him, quotes from others as well. On the postulates and axioms the scholia give more than is found in Proclus. Hultsch conjectures that the scholium on Book V., No. 3, attributing the discovery of the theorems to Eudoxus but their arrangement to Euclid, represents the tradition going back to Geminus, and that the scholium XIII., No. 1, has the same origin. A word should be added about the numerical illustrations of Euclid's propositions in the scholia to Book x. They contain a large number of calculations with sexagesimal fractions'; the fractions go as far as fourth-sixtieths (i/6o*). Numbers expressed in these fractions are handled with skill and include some results of surprising accuracy* 1 Art. " Eukleides" in Pauly-Wissowa's KeaJ-Ettcyflopadit. 1 Hultsch has written upon these in Biblietluea Matktmatisa, v a , 1904, pp. 155 — 133. * Thus v'W) is given (allowing for a slight correction by means of the context) as 5 1 i 1 46" 10'", which gives for V3 the value t 43 Jj" 13"', being the same value as that given by Hipparchus in his Table of Chords, and correct to the seventh decimal place. Similarly J 8 is given as 1 49' 41" 10'" 10"", which is equivalent to \'i= IV 141 133s- Hultsch gives instances of the various operations, addition, subtraction, etc., carried out in these fractions, and shows how the extraction of the square root was effected. Cf. T- L. Heath, History ttj Creek Mathematics, t. , pp. 50 — 03- CHAPTER VII. EUCLID IN ARABIA, We are told by Hajl Khalfa' that the Caliph al-MansQr (754-775) sent a mission to the Byzantine Emperor as the result of which he obtained from him a copy of Euclid among other Greek books, and again that the Caliph al-Ma'mun (813-833) obtained manuscripts of Euclid, among others, from the Byzantines. The version of the Elements by al-Hajjaj b. Yusuf b. Matar is, if not the very first, at least one of the first books translated from the Greek into Arabic'. According to the Fikrist* it was translated by al-Hajjaj twice ; the first translation was known as " Haruni" (" for Harun"), the second bore the name "Ma'muni" ("for al-Ma'mun") and was the more trust- worthy. Six Books of the second of these versions survive in a Leiden MS. (Codex Leidensis 399, 1) now in part published by Besthorn and Heiberg*. In the preface to this ms. it is stated that, in the reign of Harun ar-Rashid (786-809), al-Hajjaj was commanded by Yahya b. Khalid b. Barmak to translate the book into Arabic. Then, when al-Ma'mun became Caliph, as he was devoted to learning, al-Hajjaj saw that he would secure the favour of al-Ma'mun "if he illustrated and expounded this book and reduced it to smaller dimensions. He accordingly left out the superfluities, filled up the gaps, corrected or removed the errors, until he had gone through the book and reduced it, when corrected and explained, to smaller dimensions, as in this copy, but without altering the substance, for the use of men endowed with ability and devoted to learning, the earlier edition being left in the hands of readers." The Fikrist goes on to say that the work was next translated by Ishaq b. Hunain, and that this translation was improved by Thabit b, Qurra. This Abu Ya'qub Ishaq b. Hunain b. Ishaq al-Tbadi (d. 910) was the son of the most famous of Arabic translators, Hunain b. Ishaq al-'lbadi (809-873), a Christian and physician to the Caliph al- Mutawakkil (847-861). There seems to be no doubt that Ishaq, who 1 ijxicon bibliegr. et tncyclop. ed. FliigeL lilt pp. 91 , pa. 1 Klamroth, Zeitschrift der Deittschen Morgenltindisehcn Gesellsihaft, XXXV. p, 303. 3 Fihrist (tr. Suter), p. 16. * Codex Leidensis 399, 1. Eueiidis Elementa ex interpretaiione at-Hadschdsehadschii eum cgmmmfariis al-Narizii* Elauniae, part C. i. 1893, part t. ii. 1897, part U- L [900, pari ][. ii. 1905, part III. i. 1910. 76 INTRODUCTION [ch. vii must have known Greek as well as his father, made his translation direct from the Greek. The revision must apparently have been the subject of an arrangement between Ishaq and Thabit, as the latter died in 901 or nine years before Ishaq, Thabit undoubtedly consulted Greek MSS. for the purposes of his revision. This is expressly stated in a marginal note to a Hebrew version of the Elements, made from Ishaq's, attributed to one of two scholars belonging to the same family, viz. either to Moses b. Tibbon (about 1 244- 1 274) or to Jakob b. Machir (who died soon after 1306) 1 . Moreover Thabit observes, on the pro- position which he gives as ix. 31, that he had not found this proposition and the one before it in the Greek but only in the Arabic ; from which statement Klamroth draws two conclusions, (1) that the Arabs had already begun to interest themselves in the authenticity of the text and (2) that Thabit did not alter the numbers of the propositions in Ishaq's translation'. The Fihrist also says that Yuhanna al-Qass (i.e. " the Priest ") had seen in the Greek copy in his possession the pro- position in Book I. which Thabit took credit for, and that this was confirmed by Nazlf, the physician, to whom Yuhanna had shown it This proposition may have been wanting in Ishaq, and Thabit may have added it, but without claiming it as his own discovery*. As a fact, t. 45 is missing in the translation by al-Hajjaj. The original version of Ishaq without the improvements by Thabit has probably not survived any more than the first of the two versions by al-Hajjaj ; the divergences between the MSS, are apparently due to the voluntary or involuntary changes of copyists, the former class varying according to the degree of mathematical knowledge possessed by the copyists and the extent to which they were influenced by considerations of practical utility for teaching purposes*. Two MSS. of the Ishaq-Thabit version exist in the Bodleian Library (No. 279 belonging to the year 1238, and No. 280 written in 1260-1) 11 ; Books I. — XIII. are in the Ishaq-Thabit version, the non- Euclidean Books XIV., XV. in the translation of Qusta b. Luqa at-Ba'labakki (d. about 912). The first of these MSS. (No. 279) is thafctX)) used by Klamroth for the purpose of his paper on the Arabian Euclid. The other MS. used by Klamroth is (K) Kjobenhavn LXXXI, undated but probably of the 13th c, containing Books v. — xv., Boiks V. — X. being in the Ishaq-Thabit version, Books XI. — XIII. purporting to be in al-Hajjaj's translation, and Books XIV, xv. in the version of Qusta b. Luqa. In not a few propositions K and O show not the slightest difference, and, even where the proofs show considerable differences, they are generally such that, by a careful comparison, it is possible to reconstruct the common archetype, so that it is fairly clear that we have in these cases, not two recensions of one translation, but arbitrarily altered and 1 Steinschneider, Ziitschrift fur Math. «. Physik, XXXI., hist. -lilt. Abtheilung, pp. 85, 86, 90. J Klamtoth, p. »79, ■ Steinschneider, p, 88. * Klamroth, p. 306. 1 These MSS. are described by Nicoll and Pusey, Catatogus tod. m$s. orient, bibl. Bca- hiattot, pt. u. 1835 (pp. 157—161). ch. vh] EUCLID IN ARABIA 77 shortened copies of one and the same recension 1 . The Bodleian MS. No. 280 contains a preface, translated by Nicoll, which cannot be by Thibit himself because it mentions Avkenna (980-1037) and other later authors. The MS. was written at Maraga in the year 1260-1 and has in the margin readings and emendations from the edition of Nasiraddln at-Tusi (shortly to be mentioned) who was living at Maraga at the time, is it possible that at-Tusl himself is the author of the preface*? Be this as it may, the preface is interesting because it throws light on the liberties which the Arabians allowed themselves to take with the text. After the observation that the book (in spite of the labours of many editors) is not free from errors, obscurities, redundancies, omissions etc., and is without certain definitions neces- sary for the proofs, it goes on to say that the man has not yet been found who could make it perfect, and next proceeds to explain (1) that Avicenna "cut out postulates and many Definitions" and attempted to clear up difficult and obscure passages, (2) that Abu'l Wafa al-Buzjanl (939-99?) "introduced unnecessary additions and left out many things of great importance and entirely necessary," inasmuch as he was too long in various places in Book VI. and too short in Book X. where he left out entirely the proofs of the apotomae, while he made an unsuccessful attempt to emend XII. 14, (3) that Abu Ja'far al-Khazin (d, between 961 and 971) arranged the postulates excellently but " disturbed the number and order of the propositions, reduced several propositions to one " etc. Next the preface describes the editor's own claims' and then ends with the sentences, " But we have kept to the order of the books and propositions in the work itself (i.e. Euclid's) except in the twelfth and thirteenth books. For we have dealt in Book xni, with the (solid) bodies and in Book XII. with the surfaces by themselves." After Thabit the Fihrist mentions Abu 'Uthman ad-Dimashql as having translated some Books of the Elements including Book X. (It is Abu 'Uthman's translation of Pappus' commentary on Book X. which Woepcke discovered at Paris.) The Fikrist adds also that " Nazif the physician told me that he had seen the tenth Book of Euclid in Greek, that it had 40 propositions more than the version in common circulation which had 109 propositions, and that he had determined to translate it into Arabic." But the third form of the Arabian Euclid actually accessible to us is the edition of AbQ Ja'far Muh. b. Muh. b. al-Hasan Nasiraddln at-Tusi (whom we shall call at-Tusi for short), born at Tus ( m Khurasan) in 1201 (d. 1274). This edition appeared in two forms, a larger and a smaller. The larger is said to survive in Florence only (Pal. 272 and 313, the latter MS. containing only six Books) ; this was published at Rome in 1 594, and, remarkably enough, some copies of 1 KlamrOth, pp. jort — S. * Steinschneider, p. 98. Heiberg has Quoted the whole of this preface in the Ztitschrift fur Math. ft, Phytik, XXIX., hist.-litt. Abth. p. 16. * This seems to include a rearrangement of the contents of Books xiv., xv. added to the EltmftUs. 78 INTRODUCTION |ch. vu this edition are to be found with 12 and some with 13 Books, some with a Latin title and some without'. But the book was printed in Arabic, so that Kastner remarks that he will say as much about it as can be said about a book which one cannot read*. The shorter form, which however, in most MSS., is in 15 Books, survives at Berlin, Munich, Oxford, British Museum (974, 1334*, 1335), Paris (2465, 2466), India Office, and Constantinople ; it was printed at Constantinople in 1 80 1, and the first six Books at Calcutta in 1824*. At-Tusi's work is however not a trattslatiott of Euclid's text, but a re-written Euclid based on the older Arabic translations. In this respect it seems to be like the Latin version of the Elements by Campanus (Campano), which was first published by Erhard Ratdolt at Venice in 1482 (the first printed edition of Euclid*). Campanus (13th c.) was a mathematician, and it is likely enough that he allowed himself the same liberty as at-TGst in reproducing Euclid. What- ever may be the relation between Campanus' version and that of Athelhard of Bath (about 1 1 20), and whether, as Curtze thinks*, they both used one and the same Latin version of 10th — 1 ith c, or whether Campanus used Athel hard's version in the same way as at- T fist used those of his predecessors 7 , it is certain that both versions came from an Arabian source, as is evident from the occurrence of Arabic words in them*. Campanus' version is not of much service for the purpose of forming a judgment on the relative authenticity of the Greek and Arabian tradition ; but it sometimes preserves traces of the purer source, as when it omits Theon's addition to vi. 33'. A curious circumstance is that, while Campanus' version agrees with at-Tusi's in the number of the propositions in all the genuine Euclidean Books except V. and IX., it agrees with At hd hard's in having 34 propositions in Book V. (as against 25 in other versions), which confirms the view that the two are not independent, and also leads, as Klamroth says, to this dilemma: either the additions to Book V. are Athelhard's own, or he used an Arabian Euclid which is not known to us". Heiberg also notes that Campanus' Books XIV., XV. show a certain agreement with the preface to the Thabit-Ishiq version, in which the author claims to have (1) given a method of inscribing spheres in the five regular solids, (2) carried further the solution of the problem how 1 Suter, Dit Afatkeniatiier und Asironomen dtr Arobtr, p. iji. The Lai in title- a Euilidii tltttteniorum geometrkorum libri trtdtcim. Ex traditions dottissimi Nasiridini Ttami nunc frimum arabite itnprtssi. Kumae in typograpbia Medicea MDXCIV. Cum licentia superiorum. 1 Kastner, Gesehiehie der Matntmatik, I. p. 367. * Suter has a note that this US. is very old, having been copied from the original in the author's lifetime. * Suter, p. iji. ■ Described by Kastner, Gathitkte dtr Mathrmatii, 1. pp. 389 — iog, and by Weiss - enbom, Die Uberutxungtn da Etttlid durch Campano und Zamberti, Halle a. S., 1881, pp. 1 — J. See also infra. Chapter vni, p, 97. ■ Sonderabdruck des Jahrtsberitklts titer die Fortiekrittt dtr klastitcktn AUtrlnumt- wisstnsehaft vem. Okt. 1S79 — 18B3, Berlin, 1884. 7 Klamroth, p. 171. * Curtw, op. tit. p. 10; Heiberg, Exklid-Studitn, p. 178. * Heiberg's Euclid, vol. v. p. ci. l0 Klamroth, pp. 173 — 4. ch. vn] EUCLID IN ARABIA 79 to inscribe any one of the solids in any other and (3) noted the cases where this could not be done 1 . With a view to arriving at what may be called a common measure of the Arabian tradition, it is necessary to compare, in the first place, the numbers of propositions in the various Books. Haji Khalfa says that al-Hajjaj's translation contained 468 propositions, and Thabit's 478 ; this is stated on the authority of at-TQsI, whose own edition contained 468*. The fact that Thabit's version had 478 propositions is confirmed by an index in the Bodleian MS. 279 (called O by Klamroth). A register at the beginning of the Codex Leiden sis 399, 1 which gives Ishaq's numbers (although the translation is that of al-FIajjaj) apparently makes the total 479 propositions (the number in Book XIV. being apparently 11, instead of the 10 of O 1 ). I subjoin a table of relative numbers taken from Klamroth, to which I have added the corresponding numbers in August's and Heiberg's editions of the Greek text The Arabian Euclid The Greek Euclid Books Ishiq at-TOsI Campaniu Gregory August Heiberg 1 48 48 48 48 48 48 11 14 14 >4 14 14 14 1U 36 36 36 37 37 37 IV 16 16 16 16 16 16 V *S n 34 *5 n *s VI 33 32 32 33 33 33 VII 39 39 39 41 41 39 VIII *7 'S IS 37 27 37 IX 3« 36 39 36 36 36 X 109 107 107 117 116 115 XI 41 4' 4< 40 40 39 XII ■5 '5 '5 iS iS 18 XIII 21 iS ■ S ■ S iS 18 462 453 464 470 469 465 [XIV 10 10 18 7 ? XV 6 6 '3 10 478 468 495 "487 a The numbers in the case of Heiberg include all propositions which he has printed in the text ; they include therefore xiii. 6 and in. 12 now to be regarded as spurious, and X. .1 12 — 115 which he brackets as doubtful. He does not number the propositions in Books XIV., XV., but I conclude that the numbers in P reach at least 9 in xiv., and 9 in XV. 1 Heiberg, Zttisthrift fur Math. u. PAjriii, xxix,, hist.-liu. Atrtheilung, p. 11. ' Klamroth, p. 17+; Steinachneider, Zatschrift fur Math. u. Physth, XXXI., hid Mitt. Abth. p. 9*i. ■ BeMhorn- Heiberg read " 11?" aa the number, Klamroth had read it as ji (p. 173). 8o INTRODUCTION [en. vii The Fihrist confirms the number 109 for Book X., from which K lam roth concludes that Ishaq's version was considered as by far the most authoritative. In the text of O, Book IV. consists of 1 7 propositions and Book XIV. of 12, differing in this respect from its own table of contents ; IV. 15, 16 in O are really two proofs of the same proposition. In al-Hajjaj's version Book I. consists of 47 propositions only, I. 45 being omitted. It has also one proposition fewer in Book III., the Heron ic proposition m. 12 being no doubt omitted. In speaking of particular propositions, I shall use Heiberg's numbering, except where otherwise stated. The difference of 10 propositions between Tha bit- Ishaq and at-TusT is accounted for thus : (1) The three propositions vi. 12 and X, 28, 39 which both Ishaq and the Greek text have are omitted in at-Tusi, (2) Ishaq divides each of the propositions xm. 1 — 3 into two, making six instead of three in at-Tusi and in the Greek. {3) Ishaq has four propositions (numbered by him vm. 24, 2$, IX. 30, 3 1) which are neither in the Greek Euclid nor in at-Tusi. Apart from the above differences al-iiajjaj (so far as we know), Ishaq and at-Tusi agree , but their Euclid shows many differences from our Greek text. These differences we will classify as follows 1 . 1. Prepositions. The Arabian Euclid omits VII. 20, 22 of Gregory's and August's editions (Heiberg, App. to VoL 11. pp. 428-32) ; vm. 16, 17; X. 7, 8, 13, 16, 24, 112, 113, 114, besides a lemma vulgo X. 13, the proposition X. 1 1 7 of Gregory's edition, and the scholium at the end of the Book (see for these Heiberg's Appendix to Vol. III. pp. 382, 408 — 416) ; XI. 38 in Gregory and August (Heiberg, App. to Vol. I v. p. 354); XII. 6, 13, 14 ; (also all but the first third of Book xv.). The Arabian Euclid makes III. 11, 12 into one proposition, and divides some propositions (X. 31, 32 ; xi, 31, 34; xin. 1 — 3) into two each. The order is also changed in the Arabic to the following extent. v. 12, 13 are interchanged and the order in Books Vi,, vn, IX. — XIII. is : VL 1—8, 13, II, 12, 9, IO, 14—17, 19, 20, 18, 21, 22, 24, 26, 23, 25. 27—30, 32, 31, 33. VII. I — 20, 22, 21, 23 — 28, 31, 32, 29, 30, 33 — 39. IX i — 13, 20, 14—19, 21 — 25, 27, 26, 28 — 36, with two new pro- positions coming before prop. 30. x. 1—6, 9—12. IS. 14. i7— 2 3. 26—28, 25, 29—30, 31, 32, 33— in, us- XL 1—30,31.32,34.33.35—39 xii. 1 — 5, 7, 9, 8, 10, 12, 11, 15, 16 — 18. xm. 1—3, 5. 4, 6, 7, 12, 9, io, 8, 11, 13, is, 14. 16—18. 1 See KJamroth, pp. 175 — 6, 180, 161 — 4, %\± — 15, 516 ; Heibeig, vol. v. pp. xcvi, xcvii. CH.vn] EUCLID IN ARABIA Si 2. Definitions. The Arabic omits the following definitions: iv. Deff. 3 — 7, VII. Def. 9 (or io), xi. Deff, 5 — 7, 15, 17. 23, 25—28; but it has the spurious definitions VI. Deff. 2, 5, and those of proportion and ordered proportion in Book V. (Deff. 8, 19 August), and wrongly interchanges v. Deff. 1 1, 12 and also vi. Deff. 3, 4. The order of the definitions is also different in Book VII. where, after Def. II, the order is 12, 14, 13, 15, 16, 19, 20, 17, 18, 21, 22, 23, and in Book xi. where the order is 1, 2, 3,4, 8, io, 9, 13, 14, 16, 12, 21, 22, 18, 19, 20, 11, 24. 3. Lemmas and porisms. All are omitted in the Arabic except the porisms to vi. 8, vin, 2, X. 3 ; but there are slight additions here and there, not found in the Greek, e.g. in vm. 14, 15 (in K). 4. Alternative proofs. These are all omitted in the Arabic, except that in X, 105, 106 they are substituted for the genuine proofs ; but one or two alternative proofs are peculiar to the Arabic (vi. 32 and vm. 4, 6). The analyses and syntheses to XIII. I — 5 are also omitted in the Arabic. K lam roth is inclined, on a consideration of all these differences, to give preference to the Arabian tradition over the Greek (1) "on historical grounds," subject to the proviso that no Greek MS. as ancient as the 8th c, is found to contradict his conclusions, which are based generally (2) on the improbability that the Arabs would have omitted so much if they had found it in their Greek MSS., it being clear from the Fihrist that the Arabs had already shown an anxiety for a pure text, and that the old translators were subjected in this matter to the check of public criticism. Against the " historical grounds," Heiberg is able to bring a considerable amount of evidence 1 . First of all there is the British Museum palimpsest (L) of the 7th or the beginning of the 8th c. This has fragments of propositions in Book X. which are omitted in the Arabic ; the numbering of one proposition, which agrees with the numbering in other Greek ms., is not comprehensible on the assumption that eight preceding propositions were omitted in it, as they are in the Arabic; and lastly, the readings in L are tolerably like those of our MSS., and surprisingly tike those of B. It is also to be noted that, although P dates from the 10th c. only, it contains, according to all appearance, an ante-Theonine recension. Moreover there is positive evidence against certain omissions by the Arabians. At-Tusi omits VI. 12, but it is scarcely possible that, if Eutocius had not had it, he would have quoted Vi. 23 by that number*. This quotation of VI. 23 by Eutocius also tells against Ishaq who has the proposition as vi. 25. Again, Simplicius quotes VI. 10 by that number, whereas it is VI. 13 in Ishaq ; and Pappus quotes, by number, XIII. 2 (Ishaq 3, 4), X.III. 4 (Ishaq 8), XIII. 16 (Ishaq 19). 1 Heiberg in Ztitxhrift fur Math. u. Pfytit, XXIX., bilt.-litt. Ablh. p. 3 sqq. 1 Apollonius, ed. Heiberg, vol. II. p. »i8, 3 — 5. 8a INTRODUCTION [en. vii On the other hand the contraction of III. II, 12 into one proposition in the Arabic tells in favour of the Arabic. Further, the omission of certain porisms in the Arabic cannot be supported; for Pappus quotes the porism to XIII. 17 1 , Procius those to II. 4, in. 1, vii. 2\ and Simplicius that to IV. 15. Lastly, some propositions omitted in the Arabic are required in later propositions. Thus X. 13 is used in X. 18, 22, 23, 26 etc. ; X. 17 is wanted in X. 18, 26, 36; xn. 6, 13 are required for XII. 1 1 and XII. 15 respectively. It must also be remembered that some of the things which were properly omitted by the Arabians are omitted or marked as doubtful in Greek MSS. also, especially in P, and others are rightly suspected for other reasons (e.g. a number of alternative proofs, lemmas, and porisms, as well as the analyses and syntheses of XII I. 1—5). On the other hand, the Arabic has certain interpolations peculiar to our inferior MSS. (cf. the definition VI. Def. 2 and those of proportion and ordered proportion), Heiberg comes to the general conclusion that, not only is the Arabic tradition not to be preferred offhand to that of the Greek mss., but it must be regarded as inferior in authority. It is a question how far the differences shown in the Arabic are due to the use of Greek MSS. differing from those which have been most used as the basis of our text, and how far to the arbitrary changes made by the Arabians themselves. Changes of order and arbitrary omissions could not surprise us, in view of the preface above quoted from the Oxford MS. of Thabit-Ishaq, with its allusion to the many important and necessary things left out by Abu '1 Wafa and to the author's, own rearrangement of Books XII., xm. But there is evidence of differences due to the use by the Arabs of other Greek Mss. Heiberg' is able to show considerable resemblances between the Arabic text and the Bologna MS. b in that part of the MS. where it diverges so remarkably from our other MSS. (see the short description of it above, p. 49) ; in illustration he gives a comparison of the proofs of XII. 7 in b and in the Arabic respectively, and points to the omission in both of the proposition given in Gregory's edition as XI 38, and to a remark- able agreement between them as regards the order of the propositions of Book XII. As above stated, the remarkable divergence of b only affects Books xi. (at end) and XII. ; and Book xm. in b shows none of the transpositions and other peculiarities of the Arabic. There are many differences between b and the Arabic, especially in the definitions of Book XX, as well as in Book xm. It is therefore a question whether the Arabians made arbitrary changes, or the Arabic form is the more ancient, and b has been altered through contact with other MSS. Heiberg points out that the Arabians must be alone responsible for their definition of a prism, which only covers a prism with a triangular base. This could not have been Euclid's own, for the word prism already has the wider meaning in Archimedes, and 1 Pippus, V. p. 436, 5. * Procius, pp. 303 — 4. * Ztttschrift fur Math. u. Physik, XXIX., hbt.-titt. Al.ib. p. 6«qq. CH, vii] EUCLID IN ARABIA 83 Euclid himself speaks of prisms with parallelograms and polygons as bases (xi. 39; XII. 10). Moreover, a Greek would not have been likely to leave out the definitions of the " Platonic " regular solids. Heiberg considers that the Arabian translator had before him a MS. which was related to b, but diverged still further from the rest of our MSS. He does not think that there is evidence of the existence of a redaction of Books I. — X. similar to that of Books XI., XII. in b ; for K I am roth observes that it is the Books on solid geometry (XI. — XIIL) which are more remarkable than the others for omissions and shorter proofs, and it is a noteworthy coincidence that it is just in these Books that we have a divergent text in b. An advantage in the Arabic version is the omission of VII. Def. 10, although, as Iambliehus had it, it may have been deliberately omitted by the Arabic translator. Another advantage is the omission of the analyses and syntheses of XI II. I — 5 ; but again these may have been omitted purposely, as were evidently a number of porisms which are really necessary. One or two remarks may be added about the Arabic versions as compared with one another, Al-Hajjaj's object seems to have been less to give a faithful reflection of the original than to write a useful and convenient mathematical text-book. One characteristic of it is the careful references to earlier propositions when their results are used. Such specific quotations of earlier propositions are rare in Euclid ; but in al-Hajjaj we find not only such phrases as "by prop, so and so," " which was proved " or " which we showed how to do in prop, so and so," but also still longer phrases. Sometimes he repeats a construction, as in I. 44 where, instead of constructing " the parallelo- gram BEFG equal to the triangle C in the angle EBG which is equal to the angle })" and placing it in a certain position, he produces AB to G, making BG equal to half DE (the base of the triangle CDE in his figure), and on GB so constructs the parallelogram BHKG by I. 42 that it is equal to the triangle CDE, and its angle GBH is equal to the given angle. Secondly, al-Hajjaj, in the arithmetical books, in the theory of proportion, in the applications of the Pythagorean 1. 47, and generally where possible, illustrates the proofs by numerical examples. It is true, observes Klamroth, that these examples are not apparently separated from the commentary of an-Nairtzi, and might not there- fore have been due to al-Hajjaj himself; but the marginal notes to the Hebrew translation in Municn MS. 36 show that these additions were in the copy of al-Hajjaj used by the translator, for they expressly give these proofs in numbers as variants taken from al-Hajjaj 1 . These characteristics, together with al-Hajjaj 's freer formulation of the propositions and expansion of the proofs, constitute an in- telligible reason why Ishaq should have undertaken a fresh translation from the Greek. Klamroth calls Ishaq's version a model of a good translation of a mathematical text ; the introductory and transitional Klamruth, p. 310 ; Steiiischneidet, pp. 85 — 6. 84 INTRODUCTION fc H - vii phrases are stereotyped and few in number, the technical terms are simply and consistently rendered, and the less formal expressions connect themselves as closely with the Greek as is consistent with intelligibility and the character of the Arabic language. Only in isolated cases does the formulation of definitions and enunciations differ to any considerable extent from the original. In general, his object seems to have been to get rid of difficulties and unevennesses in the Greek text by neat devices, while at the same time giving a faithful reproduction of it 1 . There are curious points of contact between the versions of al-Hajjaj and Thabit-Ishaq. For example, the definitions and enunciations of propositions are often word for word the same. Presumably this is owing to the fact that Ishaq found these de- finitions and enunciations already established in the schools in his time, where they would no doubt be leamt by heart, and refrained from translating them afresh, merely adopting the older version with some changes'. Secondly, there is remarkable agreement between the Arabic versions as regards the figures, which show considerable variations from the figures of the Greek text, especially as regards the letters ; this is also probably to be explained in the same way, all the later translators having most likely borrowed al-Hajjaj's adaptation of the Greek figures'. Lastly, it is remarkable that the version of Books XI. — XIII. in the KjfSbenhavn MS. (K), purporting to be by al-Hajjaj, is almost exactly the same as the Thabit-Ishaq version of the same Books in O. Klamroth conjectures that Ishaq may not have translated the Books on solid geometry at all, and that Thabit took them from al-Hajjaj, only making some changes in order to fit them to the translation of Ishaq'. From the facts (l) that at-TusI's edition had the same number of propositions (468) as al-Hajjaj's version, while Thabit- 1 shaq's had 478, and (2) that at-Tusl has the same careful references to earlier propositions, Klamroth concludes that at-Tusi deliberately preferred al-Hajjaj's version to that of Ishaq', Heiberg, however,, points out (1) that at-Tusi left out VI. 12 which, if we may judge by Klamroth's silence, al-Hajjaj had, and (2) al-Hajjaj's version had one proposition less in Books 1. and in. than at-Tusl has. Besides, in a passage quoted by Hajl Khalfa' from at-Tusi, the latter says that "he separated the things which, in the approved editions, were taken from the archetype from the things which had been added thereto," indicating that he had compiled his edition from both the earlier translations'. There were a large number of Arabian commentaries on, or reproductions of, the Elements or portions thereof, which will be 1 Klamroth, p. 390, illustrates Ish&q's method by his way of distinguishing l^op^dftur (to be congruent with) and lipapubfadai (to be applied to), the confusion of which by trans- lators was animadverted on by Savile. Ishiq avoided the confusion by using two entirely different words. ' Klamroth, pp. JIO— I. ' ibid. p. 187, • ibid. pp. 304—5. * ibid- p. "a 74. • Hail Khalfa, I. p. 383. 7 Heiberg, Zritschrift ftir Math. u.Phjrsik, XXIX., hist.-litt. Abth. pp.3, 3. en. vn] EUCLID IN ARABIA 85 found fully noticed by Steinschn eider 1 . I shall mention here the commentators etc. referred to in the Fikrist, with a few others. 1. Abu '1 'Abbas al-Fadl b. Hatim an-Nairlzi (born at Nairiz, died about 922) has already been mentioned'. His commentary survives, as regards Books 1. — VI., in the Codex Leidensis 399, 1, now edited, as to four Books, by Besthorn and Heiberg, and as regards Books I. — x. in the Latin translation made by Gherard of Cremona in -the 12th c. and now published by Curtze from a Cracow MS.* Its importance lies mainly in the quotations from Heron and Simplicius. 2. Ahmad b. 'Umar al-KarablsI (date uncertain, probably 9th — 10th c), " who was among the most distinguished geometers and arithmeticians'." 3. A 1-' Abbas b. Sa'ld al-Jauhan (fl. 830) was one of the astro- nomical observers under al-Ma'mun, but devoted himself mostly to geometry. He wrote a commentary to the whole of the Elements, from the beginning to the end ; also the " Book of the propositions which he added to the first book of Euclid*." 4. Muh. >. 'Isa Abu 'Abdallah al-Mahan! (d. between 874 and 884) wrotej according to the Fikrist, (1) a commentary on Eucl. Book v., (2) "On proportion," (3) "On the 26 propositions of the first Book of Euclid which are proved without reductio ad absurdum*." The work " On proportion " survives and is probably identical with, or part of, the commentary on Book V. 7 He also wrote, what is not mentioned by the Fihrist, a commentary on Eucl. Book X., a fragment of which survives in a Paris MS." 5. Abu Ja'far al- Khazin (i.e. " the treasurer " or " librarian "), one of the first mathematicians and astronomers of his time, was born in Khurasan and died between the years 961 and 971. The Fikrist speaks of him as having written a commentary on the whole of the Elements*, but only the commentary on the beginning of Book X. survives (in Leiden, Berlin and Paris) ; therefore either the notes on the rest of the Books have perished, or the Fihrist is in error 10 . The latter would seem more probable, for, at the end of his commentary, al-Khazin remarks that the rest had already been commented on by Sulaiman b. 'Usma (Leiden MS.) 11 or "Oqba (Surer), to be mentioned below. At-Khazin's method is criticised unfavourably in the preface to the Oxford MS. quoted by Nicoll (see p. 77 above). 6. Abu '1 WaiS al-Buzjanl (940-997), one of the greatest Arabian mathematicians, wrote a commentary on the Elements, but 1 Steinschneider, Zeiischriftfiir Mali. u. Physii, XXXI., hist.-litt. Abth, pp. 86 sqq. 1 Steinschneider, p. 86, Fihrist (Ir. Suter), pp. [6, 67 ; Suter, Dit Maiktmaliktr ttnd Astrmniien Jrr Arabcr (1900), p. +J. ' SuppitHUHtum ad Eudiftis Optra omnia, ed. Heiberg and Mtnge, Leipzig, 1899. 1 Fikrist , pp. 16, 38 [ Sleinschneider, p. 87 ; Suter, p. 6j. * Fihrist, pp. 16, 15; Steinschneider, p. gg . Suter, p. 12. * FUrist, pp. 16, ii, 58. 7 Suter, p. 16, note, quotes the Para MS. 1467, 16 s containing the work "on proportion" u the authority for this conjecture. 1 MS. i+S7, 39° (ef. Woepcke in Mhn. pr4s. a raead. dit sriemes, XI v., i8j6\ p. 669). * Fikrist, p. 17. ™ Suter, p. j8, note b. " Steintchneider, p. 89. 86 INTRODUCTION [ch. vii did not complete it 1 . His method is also unfavourably regarded in the same preface to the Oxford MS. 28a According to Haji Khalfa, he also wrote a book on geometrical constructions, in thirteen chapters. Apparently a book answering to this description was compiled by a gifted pupil from lectures by Abu '1 Wafa, and a Paris MS. (Anc. fonds 169) contains a Persian translation of this work, not that of Abu '1 Wafa himself. An analysis of the work was given by Woepcke*, and some particulars will be found in Cantor'. Abu '1 Wafa also wrote a commentary on Diophantus, as well as a separate "book of proofs to the propositions which Diophantus used in his book and to what he (Abu '1 Wafa) employed in his commentary*." 7. Ibn Rahawaihi al-Arjanl also commented on Eucl. Book X.* 8. 'All b, Ahmad Abu '1-Qasim al- AntakI (d. 987) wrote a commentary on the whole book 1 ; part of it seems to survive (from the 5th Book onwards) at Oxford (Catal. MSS. orient. II. 28 1) 7 . 9. Sind b. 'AH Abu 't-Taiyib was a Jew who went over to Islam in the time of al-Ma'mun, and was received among his astro- nomical observers, whose head ht became* (about 830); he died after 864. He wrote a commentary on the whole of the Elements ; " Abu 'All saw nine books of it, and a part of the tenth 8 ." His book " On the Apotomae and the Medials," mentioned by the Fihrist, may be the same as, or part of, his commentary on Book x. 10. Abu Yusuf Ya'qQb b, Muh. ar-Razi "wrote a commentary on Book X., and that an excellent one, at the instance of Ibn al- •Amid"." 11. The Fihrist next mentions al-Kindi (Abu Yusuf Ya'qub b. Ishaq b. as-Sabbah al-Kindi, d. about 873), as the author (1) of a work * on the objects of Euclid's book," in which occurs the statement that the Elements were originally written by Apollonius, the carpenter (see above, p. 5 and note), (2) of a book "on the improvement of Euclid's work," and (3) of another "on the improvement of the 14th and 15th Books of Euclid." "He was the most distinguished man of his time, and stood alone in the knowledge of the old sciences collectively ; he was called ' the philosopher of the Arabians ' ; his writings treat of the most different branches of knowledge, as logic, philosophy, geometry, calculation, arithmetic, music, astronomy and others"." Among the other geometrical works of al-Kindi mentioned by the Fihrist 1 * are treatises on the closer investigation of the results of Archimedes concerning the measure of the diameter of a circle in terms of its circumference, on the construction of the figure of the two mean proportionals, on the approximate determination of the chords ^r. v. T. v. pp. 118 — 156 and 309— $$$. * Fihrist, p. 17 ; Suter, p. ij. ■ Fihrist, p. 17. 7 Suter, p, 64. • Fikrisl, p. 17, 19 j Suter, pp. 13, i+. » Fihrist, p. r;. " Fihrist, p. 17; Suter, p. 66. " Fihrist, p. 17, 10 — [J. u The mere catalogue of al-KindT's works on the various branches of science takes up four octavo pages {1 1— ij) of Suter's translation of the Fihrist. I ch. vii] EUCLID IN ARABIA 87 of the circle, on the approximate determination of the chord (side) of the nonagon, on the division of triangles and quadrilaterals and con- structions for that purpose, on the manner of construction of a circle which is equal to the surface of a given cylinder, on the division of the circle, in three chapters etc. 12. The physician Nazif b. Yumn (or Yaman) al-Qass ("the priest ") is mentioned by the Fikrist as having seen a Greek copy of Eucl. Book X. which had 40 more propositions than that which was in general circulation (containing 109), and having determined to translate it into Arabic 1 . Fragments of such a translation exist at Paris, Nos. 18 and 34. of the MS. 24s 7 (952, 2 Suppl. Arab, in Woepcke's tract); No. 18 contains "additions to some propositions of the 10th Book, existing in the Greek language'." Nazif must have died about 990*, 13. YGhanna b. Yusuf b. al-Harith b. al-Bitriq al-Qass (d. about 980) lectures' on the Elements and other geometrical books, made translations from the Greek, and wrote a tract on the " proof" of the case of two straight lines both meeting a third and making with it, on one side, two angles together less than two right angles*. Nothing of his appears to survive, except that a tract " on rational and irrational magnitudes," No. 48 in the Paris MS. just mentioned, is attributed to him. 14. Abu Muh. al- Hasan b. 'Ubaidallah b. Sulatman b. Wahb (d. 901) was a geometer of distinction, who wrote works under the two distinct titles " A commentary on the difficult parts of the work of Euclid " and " The Book on Proportion'." Suter thinks that an- other reading is possible in the case of the second title, and that it may refer to the Euclidean work " on the divisions (of figures)*." 15. Qusta b. Luqi al-Ba'labakkl (d. about 912), a physician, philosopher, astronomer, mathematician and translator, wrote " on the difficult passages of Euclid's book" and "on the solution of arith- metical problems from the third book of Euclid 7 "; also an "intro- duction to geometry," in the form of question and answer". 16. Thabit b. Qurra (826-90 1), besides translating some parts of Archimedes and Books V. — VII. of the Conks of Apollonius, and revising Ishaq's translation of Euclid's Elements, also revised the trans- lation of the Data by the same Ishaq and the book On divisions of figures translated by an anonymous writer. We are told also that he wrote the following works : (1) On the Premisses (Axioms, Postulates etc.) of Euclid, (2) On the Propositions of Euclid, (3) On the propositions and questions which arise when two straight lines are cut by a third (or on the "proof" of Euclid's famous postulate). The last tract is extant in the MS. discovered by Woepcke (Paris 2 457> 3*°)- He is also credited with "an excellent work" in the shape of an " Introduction to the Book of Euclid," a treatise on 1 Fihriit, pp. 16, 1 j. * Woepcke, Mtm. pris. i I'acad. da scittua, XIV. pp. 666, 668. * Suter, p. 68. * FiAmt, p. 38 ; Suter. p. 60. 1 Fikrist, p. 16, rad Suter' 5 Dote, p. 60. * Suter, p. tit, note 13, ' Fikritt, p. 43, * Fihriit, p. +3 ; Soter, p. 41. 88 INTRODUCTION [ch. vii Geometry dedicated to Ismail b, Bulbul, a Compendium of Geometry, and a large number of other works for the titles of which reference may be made to Suter, who also gives particulars as to which are extant 1 . 17. Abu Sa'ld Sinan b. T habit b, Qurra, the son of the translator of Euclid, followed in his father's footsteps as geometer, astronomer and physician. He wrote an "improvement of the book of on the Elements of Geometry, in which he made various additions to the original." It is natural to conjecture that Euclid is the name missing in this description (by Ibn abl Usaibi'a); Casiri has the name Aqaton 1 . The latest editor of the Ta'rikk al-Hukamd, however, makes the name to be Iflaton (= Plato), and he refers to the statement by the Fikrist and Ibn al-Qiftl attributing to Plato a work on the Elements of Geometry translated by Qusta. It is just possible, therefore, that at the time of Qusta the Arabs were acquainted with a book on the Elements of Geometry translated from the Greek, which they attri- buted to Plato*. Sinan died in 94.3. 18. Abu Sahl Wijan (or Waijan) b. Rustam al-Kuhi (ft 988), born at Kuh in Tabaristan, a distinguished geometer and astronomer, wrote, according to the Fikrist, a " Book of the Elements" after that of Euclid*; the 1st and 2nd Books survive at Cairo, and a part of the 3rd Book at Berlin (5922)'. He wrote also a number of other geometrical works : Additions to the 2nd Book of Archimedes on the Sphere and Cylinder (extant at Paris, at Leiden, and in the India Office), On the finding of the side of a heptagon in a circle (India Office and Cairo), On two mean proportionals (India Office), which last may be only a part of the Additions to Archimedes' On the Sphere and Cylinder, etc. 19. Abu Nasr Muh. b. Muh. b. Tarkhan b. Uzlag al-Farabl (870-950) wrote a commentary on the difficulties of the introductory matter to Books I. and V. s This appears £0 survive in the Hebrew translation which is, with probability, attributed to Moses b. Tibbon'. 20. Abu 'All al-Hasan b. al- Hasan b, al-Haitham (about 965- 1039), known by the name Ibn al-Haitham or Abu 'AHal-Basri, was a man of great powers and knowledge, and no one of his time approached him in the field of mathematical science. He wrote several works on Euclid the titles of which, as translated by Woepcke from Usaibi'a, are as follows 5 : 1. Commentary and abridgment of the Etements. 2. Collection of the Elements of Geometry and Arithmetic, drawn from the treatises of Euclid and Apollonius. 3. Collection of the Elements of the Calculus deduced from the principles laid down by Euclid in his Elements. 1 Suter, pp. 34—8. 1 Fikritt (ed. Suter), p. 59, note 131 ; Suter, p. ji, note b. * See Suter in Bibliathtca Mathematical Jv a> 1903-4, pp. 396 — 7, review of JuIluk Lippert's Ihn al-Qifti. Ta^rith al-hiikamd, Leipzig, 1903. ■ Fikritt, p. 40. • Suter, p. <n, ' Suter, p. 55. ' Steinschneider, p. 91. ■ Steinscbneider, pp. 91 — 3. CK. vn] EUCLID IN ARABIA 89 4. Treatise on " measure " after the manner of Euclid's Elements, 5. Memoir on the solution of the difficulties in Book I. 6. Memoir for the solution of a doubt about Euclid, relative to Book v, 7. Memoir on the solution of a doubt about the stereometric portion. 8. Memoir on the solution of a doubt about Book XII. 9. Memoir on the division of the two magnitudes mentioned in X. r (the theorem of exhaustion). 10. Commentary on the definitions in the work of Euclid (where Steinschneider thinks that some more general expression should be substituted for "definitions"). The last-named work (which Suter calls a commentary on the Postulates of Euclid) survives in an Oxford ms. (Catal. MSS. orient. I. 908) and in Algiers (1446, i"). A Leiden MS. (966) contains his Commentary "on the difficult places " up to Book v. We do not know whether in this commentary, which the author intended to form, with the commentary on the Musadarat, a sort of complete commentary, he had collected the separate memoirs on certain doubts and difficult passages mentioned in the above list A commentary on Book V. and following Books found in a Bodleian MS. (Catal. II. p. 262) with the title " Commentary on Euclid and solution of his difficulties " is attributed to b. Haitham ; this might be a continuation of the Leiden US. The memoir on X. 1 appears to survive at St Petersburg, MS. de l'lnstitut des langues orient. 192, 5° (Rosen, Catal. p. 125). 21. Ibn Sina, known as Avicenna (980-1037), wrote a Com- pendium of Euclid, preserved in a Leiden MS. No. 1445, and forming the geometrical portion of an encyclopaedic work embracing Logic, Mathematics, Physics and Metaphysics 1 . 22. Ahmad b. al-Husain al-Ahwazt al-Katib wrote a com-: mentary on Book X., a fragment of which (some 10 pages) is to be found at Leiden (970), Berlin (5923) and Paris (2467, 1 8°)*. 25. Naslraddln at-TusT (1 201-1274) who, as we have seen, brought out a Euclid in two forms, wrote : 1. A treatise on the postulates of Euclid (Paris, 2467, 5 ). 2. A treatise on the 5th postulate, perhaps only a part of the foregoing (Berlin, 5942, Paris, 2467, 6°). 3. Principles of Geometry taken from Euclid, perhaps identical with No. I above (Florence, Pal. 298). 4. 105 problems out of the Elements (Cairo). He also edited the Data (Berlin, Florence, Oxford etc.)*, 24. Muh. b. Ashraf Shamsaddin as-Samarqandi (fl. 1276) wrote "Fundamental Propositions, being elucidations of 35 selected proposi- ' Sleinschneider, p. 91 ; Suter, p. 89. - Suter, p. 57. ' Suter, pp. ijo — i. go INTRODUCTION [ch. vii tions of the first Books of Euclid," which are extant at Gotha (1496 and 1497), Oxford (Catal. I. 967, 2% and Brit Mus. 1 . 25. Musa b. Muh. b. Mahmud, known as Qadizade ar-Rumi (i.e. the son of the judge from Asia Minor), who died between 1436 and 1446, wrote a commentary on the "Fundamental Propositions" just mentioned, of which many MSS. are extant 1 . It contained biographical statements about Euclid alluded to above (p. 5. note), 26. Abu Da'Gd Sulaiman b. 'Uqba, a contemporary of al-Khazin (see above, No. 5), wrote a commentary on the second half of Book X., which is, at least partly, extant at Leiden (974) under the title "On the binomials and apotomae found in the loth Book of Euclid*." 27. The Codex Leidensis 399, 1 containing al-yajjaj's transla- tion of Books I. — VL is said to contain glosses to it by Sa id b Mas'ud b. al-Qass, apparently identical with Abu Nasr Gars al-Na'ma, son of the physician Mas'ud b. al-Qass al-Bagdadl, who lived in the time of the last Caliph al-Musta'sim {d. 1258)'. 28. Abu Muhammad b. AbdalbaqT al-Bagdadl al-Faradl (d. 1141, at the age of over 70 years) is stated in the Ta'rikk al-Hukama to have written an excellent commentary on Book X. of the Elements, in which he gave numerical examples of the propositions'. This is published in Curtze's edition of an-Nairizi where it occupies pages 252 — 386". 29. Yahya b. Muh b. 'Abdan b. 'Abdalwahid, known by the name of Ibn al-Lubudi (1 210-1268), wrote a Compendium of Euclid, and a short presentation of the postulates*. 30. Abu 'Abdallah Muh. b. Mu'adh al-Jayyanl wrote a com- mentary on Eucl. Book V. which survives at Algiers (1446, 3°)'. 31. Abu Nasr Mansur b. 'All b. 'Iraq wrote, at the instance of Muh. b. Ahmad Abu 'r-Raihan al-Blruni (973-1048), a tract "on a doubtful (difficult) passage in Eucl. Book XIII." (Berlin, 5925)*, 1 Suter, p. IJ7. ' ibid. p. 175. * ibid. p. $6. 1 ibid. pp. 153— 4, nj. * Gartx, p. 14 ; Steinschneider, pp- 94 — 5. * Suter in Bibliothtca Mathtmatua, IV,, 19031 PP' »£* »9S 1 Suter has also an article an its contents, Biblintheta Mathtmatica, VII,, 1906-7, pp. 134 — ajl. * Steinsctaneider, p. 94 ; Suter, p. 146. * Suter, Nacktrige und Btrichtigurtgen, in Abk&ndlungt* »*' Goth, dtr matk, JVisstn- Khafltn, xiv., 1901, p. 170, * Suter, p. 8], and Nathtrage, p. 173. CHAPTER VIII. PRINCIPAL TRANSLATIONS AND EDITIONS OF THE ELEMENTS. Cicero is the first Latin author to mention Euclid 1 ; but it is not likely that in Cicero's time Euclid had been translated into Latin or was studied to any considerable extent by the Romans ; for, as Cicero says in another place ', while geometry was held in high honour among the Greeks, so that nothing was more brilliant than their mathematicians, the Romans limited its scope by having regard only to its utility for measurements and calculations. How very little theoretical geometry satisfied the Roman agrimensores is evidenced by the work of Balbus de mensuris', where some of the definitions of Eucl. Book i, are given. Again, the extracts from the Elements found in the fragment attributed to Censorinus (fl. 238 A.D )• are confined to the definitions, postulates, and common notions. But by degrees the Elements passed even among the Romans into the curriculum of a liberal education ; for Martianus Capella speaks of the effect of the enunciation of the proposition " how to construct an equilateral triangle on a given straight line " among a company of philosophers, who, recognising the first proposition of the Elements, straightway break out into encomiums on Euclid". But the Elements were then (c. 470 A.D.) doubtless read in Greek ; for what Martianus Capella gives* was drawn from a Greek source, as is shown by the occurrence of Greek words and by the wrong translation of I. def. 1 (" punctum vero est cuius pars nihil est"). Martianus may, it is true, have quoted, not from Euclid himself, but from Heron or some other ancient source. But it is clear from a certain palimpsest at Verona that some scholar had already attempted to translate the Elements into Latin. This palimpsest 7 has part of the " Moral reflections on the Book of Job " by Pope Gregory the Great written in a hand of the 9th c. above certain fragments which in the opinion of the best judges date from the 4th c. Among these are fragments of Vergil and of Livy, as well as a geometrical fragment which purports to be taken from the I4tn and 1 5th Books of Euclid, As a matter of fact it is from Books XIL and XIII. and is of the nature of a free rendering, or rather a new 1 Di tratort III. 131. ' Tusc. I. j. * Gromalici Vtttra, 1. 97 sij. (eri. F. Blume, K. Lachmann and A. Rudorff, Bet] in, 1848, 1851). * Censorinus, ed. Hultsch, pp. 60—3. * Martianus Capella, VI. 714. ■ ibid. VI, 708 sq. 7 Cf. Canlot, ],, p. s6j. 93 INTRODUCTION Jen. vm arrangement, of Euclid with the propositions in different order 1 . The mh. was evidently the translator's own copy, because some words are struck out and replaced by synonyms. We do not know whether the translator completed the translation of the whole, or in what relation his version stood to our other sources. Magnus Aurelius Cassiodorus (b. about 475 A.D.) in the geometrical part of his encyclopaedia De artibus ac disciplines liberalium literarum says that geometry was represented among the Greeks by Euclid, Apollonius, Archimedes, and others, " of whom Euclid was given us translated into the Latin language by the same great man Boethius ." ; also in his collection of letters* is a letter from Theodoric to Boethius containing the words, "for in your translations ... Nicomachus the arithmetician, and Euclid the geometer, are heard in the Ausonian tongue." The so-called Geometry of Boethius which has come down to us by no means constitutes a translation of Euclid. The MSS. variously give five, four, three or two Books, but they represent only two distinct compilations, one normally in five Books and the other in two. Even the latter, which was edited by Friedlein, is not genuine*, but appears to have been put together in the 11 th c, from various sources. It begins with the definitions of Eucl. I., and in these are traces of perfectly correct readings which are not found even in the MSS. of the 10th c, but which can be traced in Prod us and other ancient sources ; then come the Postulates (five only), the Axioms (three only), and after these some definitions of Eucl. 1 1., ill., IV. Next come the enunciations of Eucl. I., of ten propositions of Book II., and of some from Books HI., IV., but always without proofs ; there follows an extraordinary passage which indicates that the author will now give something of his own in elucidation of Euclid, though what follows is a literal translation of the proofs of Eucl. I. 1 — 3. This latter passage, although it affords a strong argument against the genuineness of this part of the work, shows that the Pseudoboethius had a Latin translation of Euclid from which he extracted the three propositions. Curtze has reproduced, in the preface to his edition of the trans- lation by Gherard of Cremona of an-Nairizt's Arabic commentary on Euclid, some interesting fragments of a translation of Euclid taken from a Munich MS. of the 10th c. They are on two leaves used for the cover of the MS. (Bibliothecae Regiae Universitatis Monacensis 2° 757) and consist of portions of Eucl. I. 37, 38 and II. 8, translated literally word for word from the Greek text. The translator seems to have been an Italian (cf. the words "capitolonono" used for the ninth prop, of Book II.) who knew very little Greek and had moreover little mathematical knowledge. For example, he translates the capital letters denoting points in figures as if they were numerals : thus to. A Br, 1 The fragment was deciphered by W. Studernund, who communicated his results to Cantor. * Cassiodorus, Variat, I. 4;, p. +0, 13 cd. Mommsen. * See especial Ijr Weissenbom in Abhandltmgen tur Gtsti. d. Math. 11. p. 185 so,.; Heiberg in Phihlogus, XLI1I. p. 507 sq. ; Cantor, 13, p. 580 so,. ch. vni] TRANSLATIONS AND EDITIONS 93 AEZ is translated "que primo secundo et tertio quarto quinto et sept i mo," T becomes "tricentissimo " and so on. The Greek MS. which he used was evidently written in uncials, for AEZ8 becomes in one place " quod autem septimo nono," showing that he mistook AE for the particle Si, and xal 6 2TU is rendered "sicut tricentissimo et quadringentissimo," showing that the letters must have been written KAIOCTU. The date of the Englishman Athelhard (^Ethelhard) is approxi- mately fixed by some remarks in his work Perdifficites Quaestiones Naturaks which, on the ground of the personal allusions they contain, must be assigned to the first thirty years of the 12th c. 1 He wrote a number of philosophical works. Little is known about his life. He is said to have studied at Tours and Lao 11, and to have lectured at the latter school. He travelled to Spain, Greece, Asia Minor and Egypt, and acquired a knowledge of Arabic, which enabled him to translate from the Arabic into Latin, among other works, the Elements of Euclid. The date of this translation must be put at about 1120. MSS. purporting to contain Atheihard's version are extant in the British Museum (Harleian No. 5404 and others), Oxford (Trin. Coll. 47 and Ball. Coll. 257 of 12th c), Niirnberg (Johannes Regiomontanus' copy) and Erfurt. Among the very numerous works of Gherard of Cremona (I 1 14 — 1 1 87) are mentioned translations of " 1 5 Books of Euclid " and of the Data*. Till recently this translation of the Elements was supposed to be lost; but Axel Anthon Bjornbo has succeeded (1904) in discovering a translation from the Arabic which is different from the two others known to us (those by Athelhard and Campanus respectively), and which he, on grounds apparently convincing, holds to be Gherard's. Already in 190 1 Bjornbo had found Books X.— XV. of this translation in a MS. at Rome (Codex Reginensis lat. 1268 of 14th c.)*; but three years later he had traced three MSS. containing the whole of the same translation at Paris (Cod. Paris. 7216, 15th a), Baulogne-sur-Mer (Cod. Bononiens. 196, 14th a), and Bruges (Cod, Brugens. 521, 14th c), and another at Oxford (Cod. Digby [74, end of 12th c.) containing a fragment, XI. 2 to XIV. The occurrence of Greek words in this translation such as rombus, romboides (where Athelhard keeps the Arabic terms), ambligonius, ortAogonius, gnomo, fyramis etc., show that the translation is independent of Atheihard's. Gherard appears to have had before him an old translation of Euclid from the Greek which Athelhard also often followed, especially in his terminology, using it however in a very different manner. Again, there are some Arabic terms, e.g. meguar for axis of rotation, which Athelhard did not use, but which is found in almost all the translations that are with certainty attributed to Gherard of Cremona; there occurs also the 1 Cautor, Guti. d. Math. („ p. 006. 1 Boncompagni, Delia vita t Mil ogcrt di Gherorih Crcmmusc, Rome, 1 851, p. 5. * Described in an »ppendix to Sludun iibir Mttuloes'' SpAarik [Abliaiidlungtn lur Gtithithtt dtr motktmaHichttt Wisumchafttn, xtv., 1901). * See Biblitthtca Matktmalka, vi„ 1903-6, pp. 141 — 6, 94 INTRODUCTION [cb. vjii expression "superficies equidistantium laterum et rcctorum angulorum," found also in Gherard's translation of an-Nairlzl, where Athelhard says "parallelogrammum rectangulum." The translation is much clearer than Athelhard's: it is neither abbreviated nor "edited" as A the! hard's appears to have been ; it is a word-for-word translation of an Arabic MS. containing a revised and critical edition of Thabit's version. It contains several notes quoted from Thabit himself ( Tftebit dixit), e.g. about alternative proofs etc. which Thabit found " in another Greek MS.," and is therefore a further testimony to Thabit's critical treatment of the text after Greek MSS. The new editor also added critical remarks of his own, e.g. on other proofs which he found in other Arabic versions, but not in the Greek : whence it is clear that he compared the Thabit version before him with other versions as care- fully as Thabit collated the Greek mss. Lastly, the new editor speaks of "Thebit qui transtulit hunc librum in arabicam linguam" and of "translatio Thebit," which may tend to confirm the statement of al-Qiftl who credited Thabit with an independent translation, and not (as the Fihrist does) with a mere improvement of the version of Ishaq b, Hunain. Gherard's translation of the Arabic commentary of an-Nairlzi on the first ten Books of the Elements was discovered by Maximilian Curtze in a MS. at Cracow and published as a supplementary volume to Heiberg and Menge's Euclid 1 : it will often be referred to in this work. Next in chronological order comes Johannes Campanus (Campano) of Novara. He is mentioned by Roger Bacon (12 14-1 294) as a prominent mathematician of his time 1 , and this indication of his date is confirmed by the fact that he was chaplain to Pope Urban IV, who was Pope from 1261 to 1281". His most important achievement was his 'edition of the Elements including the two Books XIV. and XV. which are not Euclid's. The sources of Athelhard's and Campanus' translations, and the relation between them, have been the subject of much discussion, which does not seem to have led as yet to any definite conclusion. Cantor (Hi, p. 91) gives references* and some particulars. It appears that there is a Ms. at Munich (Cod. lat. Mon. 13021} written by Sigboto in the 12th c. at Priifning near Regensburg, and denoted by Curtze by the letter R, which contains the enunciations of part of Euclid. The Munich MSS. of Athelhard and Campanus' translations have many enunciations textually identical with those in R, so that the source of all three must, for these enunciations, have 1 Anaritii in decern libros prism Etemenierum Eucfidis Comnuntarii ex interpretation* Gherardi Cremenensis in codicc Cracaviensi 569 strvata edidit M ax i mi Nanus Curtze, Leipzig (Teubner), 1899. * Cantor, iij, p. 88. * Tiraboschi, Storia delta tetteratura italiana, IV, 145 — 160, 4 H. Weissenborn in Zeitichrift fiir Math. u. Physii, XXV., Supplement, pp. i+j— 166, and in his monograph. Die Ubersttiungen des Euktid dureh Campano und Zamberti (188?); Max. Curtze in Philelogisc&e Rundschau (iSSi), I. pp. $13—950, and in Jakrtsherieht iiitr die Fertschritti det tlaaischen Alterthumfwissemchafl t XL. (188+, in.) pp. 19 — »' ; Heiberg in ZeitscArift fiir Math. u. Pfyrik, xxxv., bist.-litt. Ablh., pp. 48 — j8 and pp. Si— 6. en. vm] TRANSLATIONS AND EDITIONS 95 been the same; in others Athelhard and Campanus diverge com- pletely from R, which in these places follows the Greek text and is therefore genuine and authoritative. In the 32nd definition occurs the word "elinuam," the Arabic term for " rhombus," and throughout the translation are a number of Arabic figures. But R was not translated from the Arabic, as is shown by (among other things) its close resemblance to the translation from Euclid given on pp. 377 sqq. of the Gromatici Ve teres and to the so-called geometry of Boethius. The explanation of the Arabic figures and the word " elinuam " in Def. 32 appears to be that R was a late copy of an earlier original with corruptions introduced in many places ; thus in Def. 32 a part of the text was completely lost and was supplied by some intelligent copyist who inserted the word "elinuam," which was known to him, and also the Arabic figures. Thus Athelhard certainly was not the first to translate Euclid into Latin ; there must have been in existence before the nth c, a Latin translation which was the common source of R, the passage in the Gromatici, and " Boethius." As in the two latter there occur the proofs as well as the enunciations of 1. 1 — 3, it is possible that this translation originally contained the proofs also. Athelhard must have had before him this translation of the enunciations, as well as the Arabic source from which he obtained his proofs. That some sort of translation, or at least fragments of one, were available before Athelhard's time even in England is indicated by some old English verses 1 : "The clerk Euclide on this wyse hit fonde Thys craft of gemetry yn Egypte londe Yn Egypte he tawghte hyt fill wyde, In dyvers londe on every syde. Mony erys after warde y understonde Yer that the Craft com ynto thys londe. Thys Craft Coin into England, as y yow say, Yn tyme of good kyng Adelstone's day," which would put the introduction of Euclid into England as far back as 924-940 A.D. We now come to the relation between Athelhard and Campanus. That their translations were not independent, as Weissenborn would have us believe, is clear from the fact that in all MSS. and editions, apart from orthographical differences and such small differences as are bound to arise when MSS. are copied by persons with some knowledge of the subject-matter, the definitions, postulates, axioms, and the 364 enunciations are word for word identical in Athelhard and Campanus; and this is the case not only where both have the same text as R but where they diverge from it. Hence it would seem that Campanus used Athelhard's translation and only developed the proofs by means of another redaction of the Arabian Euclid. It is true that the difference between the proofs of the propositions in the two translations is considerable; Athelhard's are short and com- 1 Quoted by HidUwdl in Kara Mathtmatua (p. j6 note) from us. Bib. Reg. Mm. Brit. 17 A. 1. f. ** — 3. 96 INTRODUCTION [ch. viii pressed, Campari us' clearer and more complete, following the Greek text more closely, though still at some distance. Further, the arrangement in the two is different; in Athelhard the proofs regularly precede the enunciations, Campanus follows the usual order. It is a question how far the differences in the proofs, and certain additions in each, are due to the two translators themselves or go back to Arabic originals. The latter supposition seems to Curtze and Cantor the more probable one. Curtze's general view of the relation of Campanus to Athelhard is to the effect that A thel hard's translation was gradually altered, from the form in which it appears in the two Erfurt MSS. described by Weissenborn, by successive copyists and commentators who had Arabic originals before them, until it took the form which Campanus gave it and in which it was published. In support of this view Curtze refers to Regiomontanus' copy of the Athelhard-Campanus translation. In Regiomontanus' own preface the title is given, and this attributes the translation to Athelhard ; but, while this copy agrees almost exactly with Athelhard in Book I., yet, in places where Campanus is more lengthy, it has similar additions, and in the later Books, especially from Book HI. onwards, agrees absolutely with Campanus; Regiomontanus, too, himself implies that, though the translation was Athelhard's, Campanus had revised it; for he has notes in the margin such as the following, "Campani est hec," "dubito an demonstret hie Campanus " etc. We come now to the printed editions of the whole or of portions of the Elements, This is not the place for a complete bibliography, such as Riccardi has attempted in his valuable memoir issued in five parts between 1 88/ and 1893, which makes a large book in itself 1 . I shall confine myself to saying something of the most noteworthy translations and editions. It will be convenient to give first the Latin translations which preceded the publication of the editio prineeps of the Greek text in 1533, next the most important editions of the Greek text itself, and after them the most important translations arranged according to date of first appearance and languages, first the Latin translations after 1533, then the Italian, German, French and English translations in order. It may be added here that the first allusion, in the West, to the Greek text as still extant is found in Boccaccio's commentary on the Divina Commedia of Dante*. Next Johannes Regiomontanus, who intended to publish the Elements after the version of Campanus, but with the latter's mistakes corrected, saw in Italy (doubtless when staying with his friend Bessarion) some Greek MSS. and noticed how far they differed from the Latin version (see a letter of his written in the year 1471 to Christian Roder of Hamburg)*. 1 Saggie di una Bibliografia Euttidta, memoria del Prof. Pietro Riccardi (Bologna, 1887, 18M8, 1890, 1693). * 1. p. 404. 8 Published in C. T. de Murr's Memorabilia BibikHhtcarum Mffrimbergmnttm, Part 1. p. 190 scjh- ch. viu] TRANSLATIONS AND EDITIONS 97 I. Latin translations prior to 1533. 1482. In this year appeared the first printed edition of Euclid, which was also the first printed mathematical book of any import- ance. This was printed at Venice by Erhard Ratdolt and contained Campanus' translation 1 , Ratdolt belonged to a family of artists at Augsburg, where he was born about 1443. Having learnt the trade of printing at home, he went in 1475 to Venice, and founded there a famous printing house which he managed for 1 1 years, after which he returned to Augsburg and continued to print important books until 1516. He is said to have died in 1528. Kastner 1 gives a short description of this first edition of Euclid and quotes the dedication to Prince Mocenigo of Venice which occupies the page opposite to the first page of text. The book has a margin of 2i inches, and in this margin are placed the figures of the propositions. Ratdolt says in his dedication that at that time, although books by ancient and modern authors were printed every day in Venice, little or nothing mathematical had appeared : a fact which he puts down to the diffi- culty involved by the figures, which no one had up to that time succeeded in printing. He adds that after much labour he had discovered a method by which figures could be produced as easily as letters*. Experts are in doubt as to the nature of Ratdolt's discovery. Was it a method of making figures up out of separate parts of figures, straight or curved lines, put together as letters are put together to make words? In a life of Joh. Gottlob Immanuel Breitkopf, a con- temporary of Kastner's own, this member of the great house of Breitkopf is credited with this particular discovery. Experts in that same house expressed the opinion that Ratdolt's figures were wood- cuts, while the letters denoting points in the figures were like the other letters in the text ; yet it was with carved wooden blocks that printing began. If Ratdolt was the first to print geometrical figures, it was not long before an emulator arose ; for in the very same year Mattheus Cordonis of Windischgratz employed woodcut mathematical figures in printing Oresme's De latitudintbus*. How eagerly the opportunity of spreading geometrical knowledge was seized upon is proved by the number of editions which followed in the next few years. Even the year 1482 saw two forms of the book, though they only differ in the first sheet Another edition came out in i486 (Vlmae, apud lo. Regeruni) and another in 149 1 {Vincentiae per 1 Curtie (An-Nairi/i, p. xiji) repioduces the heading of the first page of the text as follows (there is no title-page) : PreclarifTimu opus elemento^t- Euclidis megaref is vna cu comentis Camparii pfpicaciffimi in arte geometria incipit iclicit 1 , after which the definitions begin at once. Other copies have the shorter heading ; Preclarissimus liber elernentorum Euclidis perspicacissimi : in artem Geometric incipit quam foeltrissime* At the end stands the following : Q Opus elcmentoru euclidis megarenf is in geometria arte Jn id quoq) Camparu pfpicacifTimi Cdmentaliones finiut. Erhardus ratdolt Augustensis impreflbr Ic-lertiiritnus . venetijs imprefTit . Anno falutis • M.cccc.Ixxxij - Octauis . Calefi . Juh . Lector . Vale. • Kastner, Gtuhhhttdtr Mathematii, i, p. J89 sqq. See also Weissenborn, DU (ffenctw- wtgen dts Euktid durtk Catttpane und Zanthtrti, pp. 1 — 7. - "Mea industria non sine maxiroo labore effect vt qua facilitate Litterarum elementa imprimuntur ea etiam geometrice figure cunficerentur. " ' Cume in Ztifschrift fur Math. u. fhysii, xx... hist.-litt. Abth. p. 58. 98 INTRODUCTION [ch. vm Leonardum de Basika et Gttlielmum de Papia), but without the dedi- cation to Mocenigo who had died in the meantime (1485). If Cam- pan us added anything of his own, his additions are at all events not distinguished by any difference of type or otherwise; the enunciations are in large type, and the rest is printed continuously in smaller type. There are no superscriptions to particular passages such as Euclides ex Campano, Campanus, Campani additio, or Campani annotatio, which are found for the first time in the Paris edition of 15 16 giving both Campanus' version and that of Zamberti (presently to be men- tioned). 1 50 1. G. Valla included in his encyclopaedic work De expetendis et fugiendis rebus published in this year at Venice (in aedibus Aldi Romani) a number of propositions with proofs and scholia translated from a Greek MS. which was once in his possession (cod, Mutin. Ill B, 4 of the 15th c.). 1505. In this year Bartolomeo Zamberti (Zambertus) brought out at Venice the first translation, from the Greek text, of the whole of the Elements. From the title 1 , as well as from his prefaces to the Catoptrica and Data, with their allusions to previous translators " who take some things out of authors, omit some, and change some," or " to that most barbarous translator " who filled a volume purporting to be Euclid's "with extraordinary scarecrows, nightmares and phantasies," one object of Zamberti's translation is clear. His animus against Campanus appears also in a number of notes, e.g. when he condemns the terms "helmuain" and " helmuariphe " used by Campanus as barbarous, un- Latin etc., and when he is roused to wrath by Campanus' unfortu- nate mistranslation of V. Def. 5. He does not seem to have had the penetration to see that Campanus was translating from an Arabic, and not from a Greek, text Zamberti tells us that he spent seven years over his translation of the thirteen Books of the Elements. As he seems to have been born in 1473, and the Elements were printed as early as 1 500, though the complete work (including the Pkaenomena, Optica, Catoptrica, Data etc.) has the date 1505 at the end, he must have translated Euclid before the age of 30. Heiberg has not been able to identify the Ms, of the Elements which Zamberti used ; but it is clear that it belonged to the worse class of mss., since it contains most of the interpolations of the Theonine variety. Zam- berti, as his title shows, attributed the proofs to Theon. 1 509. As a counterblast to Zamberti, Luca Faciuolo brought out an edition of Euclid, apparently at the expense of Ratdolt, at Venice {per Paganinunt de Pagattinis), in which he set himself to vindicate Campanus. The title-page of this now very rare edition' begins thus : "The works of Euclid of Megara, a most acute philosopher and without 1 The title begins thus: "Euclidis megaresis philosophi platonicj matbenuticanun dLscipLinaruin Janitoris : II a bent in hoc volumine quicunque ad mathematicam aubstantiam aspirant ; elemenLorum libros xiij cum exposition Theonis io&ignis mathematici. qui bos multa quae deerant ex lectione graeca sumpla addLta sunt nee non plurima perueru et praepostere : vol u la in Campani interpretatione ; ordinate digesta et castigata sunt etc." For a description of the book see Weissenborn, p. 12 fiqq- tt See Weissenborn, p. 30 sqq. CB. vih] TRANSLATIONS AND EDITIONS 99 question the chief of all mathematicians, translated by Campanus their most faithful interpreter'' It proceeds to say that the translation had been, through the fault of copyists, so spoiled and deformed that it could scarcely be recognised as Euclid, Luca Faciuolo accordingly has polished and emended it with the most critical judgment, has corrected 129 figures wrongly drawn and added others, besides supply- ing short explanations of difficult passages. It is added that Scipio Vegius of Milan, distinguished for his knowledge " of both languages" (i.e. of course Latin and Greek), as well as in medicine and the more sublime studies, had helped to make the edition more perfect. Though Zamberti is not once mentioned, this latter remark must have refer- ence to Zamberti's statement that his translation was from the Greek text ; and no doubt Zamberti is aimed at in the wish of Paciuolo's "that others too would seek to acquire knowledge instead of merely showing off, or that they would not try to make a market of the things of which they are ignorant, as it were (selling) smoke'." Weissenborn observes that, while there are many trivialities in Paci- uolo's notes, they contain some useful and practical hints and explana- tions of terms, besides some new proofs which of course are not difficult if one takes the liberty, as Paciuolo does, of diverging from Euclid's order and assuming for the proof of a proposition results not arrived at till later. Two not inapt terms are used in this edition to describe the figures of HI. 7, 8, the former of which is called the goose's foot (pes anseris), the second the peacock's tail (cauda pavonis) Paciuolo as the castigator of Campanus' translation, as he calls himself, failed to correct the mistranslation of v, Def. 5 s , Before the fifth Book he inserted a discourse which he gave at Venice on the 15th August, 1508, in S. Bartholomew's Church, before a select audience of 500, as an introduction to his elucidation of that Book. 1516. The first of the editions giving Campanus' and Zamberti's translations in conjunction was brought out at Paris (in officina Henrici Stephani e regiene scholae Decretorum). The idea that only the enun- ciations were Euclid's, and that Campanus was the author of the proofs in his translation, while Theon was the author of the proofs in the Greek text, reappears in the title of this edition ; and the enunciations of the added Books XIV., XV. are also attributed to Euclid, Hypsicles being credited with the proofs 1 . The date is not on the title-page nor at the ] "Atquc utinam et alii cognoscere vellent nun ostentare out st quae nesciunt veluti fumum venditare non conarentur." * Campanus 1 translation- in Rat dolt's edition is. as follows: "Quanti rates quae dicuntur continuom habere proportionalilatem, sunt, quarum equ£ multiplicia apt equa sunt aut eque sibi sine interruptions addunt aut minuunt" (!), to which Campanus adds the note: " Com mil e proportion alia. sunt quorum omnia multiplicia equaliasunt continue proportional ia. Sed noluit ipsam diffinitionem proponete sub mac forma, quia tunc diffipiret idem per idem, apcrte (? a parte) tamen rei est istud cum sua difhnitione converiibile. " * "Euclidis Megarensis" Geometricorum Elementorum Libri XV. Campani Galli trans* alpini in eosdem commentariorum libri XV. Theonis Alexandrini Barthoiom&eo Zamberto Veneto inlerprete, in tredecim priores, commentationum libri XI 11. Hypsiclis Alexandria! in duos posteriores, eodem Barthoiomaco Zamberto Veneto interprete, commentariorum libri II." On the last page (.6t) is a similar statement of content, but wish the difference that the expression "ex Campani..,deinde Theon is... et Hypsiclis.../rWi'fra»ft&*/." For description see Weissentarn, p. 56 sqq- ioo INTRODUCTION [ch. vih end, but the letter of dedication to Francois Brieonnet by Jacques Leftvre is dated the day after the Epiphany, 15 16. The figures are in the margin. The arrangement of the propositions is as follows : first the enunciation with the heading Eudides ex Catnpano, then the proof with the note Campanus, and after that, as Campani additio, any passage found in the edition of Campanus' translation but not in the Greek text ; then follows the text of the enunciation translated from the Greek with the heading Euclides ex Zamberto, and lastly the proof headed Theo ex Zamberto. There are separate figures for the two proofs, This edition was reissued with few changes in 1537 and 1546 at Basel (apud Jehannem Heruagium), but with the addition of the Pkaenomena, Optica, Catoptrica etc. For the edition of 1537 the Paris edition of 15 16 was collated with "a Greek copy" (as the preface says) by Christian Herlin, professor of mathematical studies at Strasshurg, who however seems to have done no more than correct one or two passages by the. help of the Basel editio princeps (1533), and add the Greek word in cases where Zamberti's translation of it seemed unsuit- able or inaccurate We now come to II. Editions of the Greek text, 1 533 is the date of the editio princeps, the title-page of which reads as follows: ETKAEIAOT STOlXEIflN BIBA>- IE> EK TON 6EON02 ZYNOYSICN. Et? tov avrov to wp&TOV, ifyyriftdT&v UpoicXov /9*/9\. 5. Adiecta praefatiuncula in qua de disciplinis Mathematicis nonnihil. BASILEAE APVD 10 AN. HERVAGIVM ANNO M.D.XXXIII. MENSE SEPTEMBRI. The editor was Simon Grynaeus the elder (d. 1541), who, after working at Vienna and Ofen, Heidelberg and Tubingen, taught last of all at Basel, where theology was his main subject. His "prae- fatiuncula " is addressed to an Englishman, Cuthbert Tonstall { 1474- ■559)i who, having studied first at Oxford, then at Cambridge, where he became Doctor of Laws, and afterwards at Padua, where in addi- tion he leamt mathematics — mostly from the works of Regiomontanus and Paciuolo— wrote a book on arithmetic 1 as "a farewell to the sciences," and then, entering politics, became Bishop of London and member of the Privy Council, and afterwards (1530) Bishop of Durham. Grynaeus tells us that he used two MSS. of the text of the Elements, entrusted to friends of his, one at Venice by "Lazarus Bayfius" (Lazare de BaJf, then the ambassador of the King of France at Venice), the other at Paris by " loann. Rvellius " (J ean Ruel, a French doctor and a Greek scholar), while the commentaries of Proclus were put at 1 Dt arte supputandi libri quatuar. ch. viu] TRANSLATIONS AND EDITIONS 101 the disposal of Grynaeus himself by " loann. Claymundus" at Oxford, Heiberg has been able to identify the two mss. used for the text ; they are (i) cod. Venetus Marcianus 301 and (2) cod. Paris, gr. 2343 of the 16th c, containing Books 1. — xv., with some scholia which are embodied in the text. When Grynaeus notes in the margin the readings from " the other copy," this " other copy " is as a rule the Paris MS., though sometimes the reading of the Paris MS. is taken into the text and the " other copy " of the margin is the Venice MS. Besides these two mss. Grynaeus consulted Zamberti, as is shown by a number of marginal notes referring to " Zampertus " or to " latin um exemplar" in certain propositions of Books IX. — XL When it is con- sidered that the two MSS. used by Grynaeus are among the worst, it is obvious how entirely unauthoritative is the text of the editio princeps. Yet it remained the source and foundation of later editions of the Greek text for a long period, the editions which followed being designed, not for the purpose of giving, from other MSS., a text more nearly representing what Euclid himself wrote, but of supplying a handy compendium to students at a moderate price. 1536. Orontius Finaeus (Oronce Fine) published at Paris {apurf Simonem Colinaeum) "demonstrations on the first six books of Euclid's elements of geometry," " in which the Greek text of Euclid himself is inserted in its proper places, with the Latin translation of Barth. Zamberti of Venice," which seems to imply that only the enunciations were given in Greek. The preface, from which Kastner quotes', says that the University of Paris at that time required, from all who aspired to the laurels of philosophy, a most solemn oath that they had attended lectures on the said first six Books. Other editions of Fine's work followed in 1544 and 1551. 1545. The enunciations of the fifteen Books were published in Greek, with an Italian translation by Angelo Caiani, at Rome (apud Antonium Bladum Asulanum). The translator claims to have cor- rected the books and " purged them of six hundred things which did not seem to savour of the almost divine genius and the perspicuity of Euclid'" 1549. Joachim Camerarius published the enunciations of the first six Books in Greek and Latin (Leipzig). The book, with preface, purports to be brought out by Rhaeticus (1514-1576), a pupil of Copernicus. Another edition with proofs of the propositions of the first three Books was published by Moritz Steinmetz in 1 577 (Leipzig) ; a note by the printer attributes the preface to Camerarius himself. 155a loan. Scheubel published at Basel (also per loan. Her- vagium) the first six Books in Greek and Latin "together with true and appropriate proofs of the propositions, without the use of letters " (i.e. letters denoting points in the figures), the various straight lines and angles being described in words 1 . 1557 (also 1558). Stephanus Gracilis published another edition (repeated 1 573, 1 578, 1 598) of the enunciations (alone) of Books I, — XV. 1 Kastner, I. p. j6o. * Heiberg, vol. v. p. crii. ' Kastner, 1. p. 359. 10a INTRODUCTION [ch. vm in Greek and Latin at Paris {apud Gulielmurn Cavellai). He remarks in the preface that for want of time he had changed scarcely anything in Books I. — VI., but ; n the remaining Books he had emended what seemed obscure or inelegant in the Latin translation, while he had adopted in its entirety the translation of Book x. by Pierre Mondore (PetrusMontaureus), published separately at Paris in 1551. Gracilis also added a few " scholia." 1564. In this year Conrad Dasypodius (Rauchfuss), the inventor and maker of the clock in Strassburg cathedral, similar to the present one, which did duty from 1571 to 1789, edited (Strassburg, Chr. Mylius) (1) Book 1. of the Elements in Greek and Latin with scholia, (2) Book 11. in Greek and Latin with Barlaam's arithmetical version of Book II., and (3) the enunciations of the remaining Books III. — XIII. Book 1. was reissued with " vocabula quaedam geometrica " of Heron, the enunciations of all the Books of the Elements, and the other works of Euclid, all in Greek and Latin. In the preface to (1) he says that it had been for twenty-six years the rule of his school that all who were promoted from the classes to public lectures should learn the first Book, and that he brought it out, because there were then no longer any copies to be had, and in order to prevent a good and fruitful regulation of his school from falling through. In the preface to the edition of 157 1 he says that the first Book was generally taught in all gymnasia and that it was prescribed in his school for the first class. In the preface to (3) he tells us that he published the enunciations of Books in.— xill. in order not to leave his work unfinished, but that, as it would be irksome to carry about the whole work of Euclid in extenso, he thought it would be more convenient to students of geometry to learn the Elements if they were compressed into a smaller book. 1620. Henry Briggs {of Briggs' logarithms) published the first six Books in Greek with a Latin translation after Commandinus, "corrected in many places" (London, G. Jones). 1703 is the date of the Oxford edition by David Gregory which, until the issue of Heiberg and Menge's edition, was still the only edition of the complete works of Euclid 1 . In the Latin translation attached to the Greek text Gregory says that he followed Comman- dinus in the main, but corrected numberless passages in it by means of the books in the Bodleian Library which belonged to Edward Bernard (1638- 1 696), formerly Savilian Professor of Astronomy, who had conceived the plan of publishing the complete works of the ancient mathematicians in fourteen volumes, of which the first was to contain Euclid's Elements I. — xv. As regards the Greek text, Gregory tells us that he consulted, as far as was necessary, not a few MSS, of the better sort, bequeathed by the great Savile to the University, as well as the corrections made by Savile in his own hand in the margin of the Basel edition. He had the help of John Hudson, Bodley's Librarian, who 1 ETKAEIAOT TA SOZOMBNA. Euclidis qnae supersunl omnia. Ex recensions Davidis Gregorii M.D. Asttonomiae Professoris Saviliani el R.S.S. Oxoniae, o Theatio Sheldoniano, An. Dora, mdccici. cm. vin] TRANSLATIONS AND EDITIONS 103 punctuated the Basel text before it went to the printer, compared the Latin version with the Greek throughout, especially in the Elements and Data, and, where they differed or inhere he suspected the Greek text, consulted the Greek msS. and put their readings in the margin if they agreed with the Latin and, if they did not agree, affixed an asterisk in order that Gregory might judge which reading was geo- metrically preferable. Hence it is clear that no Greek MS., but the Basel edition, was the foundation of Gregory's text, and that Greek MSS. were only referred to in the special passages to which Hudson called attention. 1 is 14-1 818. A most important step towards a good Greek text was taken by F. Peyrard, who published at Paris, between these years, in three volumes, the Elements and Data in Greek, Latin and French 1 . At the time (1808) when Napoleon was having valuable MSS. selected from Italian libraries and sent to Paris, Peyrard managed to get two ancient Vatican MSS. (190 and 1038) sent to Paris for his use (Vat. 304 was a ' so at Paris at the time, but all three were restored to their owners in 1 8 14). Peyrard noticed the excellence of Cod. Vat. 190, adopted many of its readings, and gave in an appendix a conspectus of these readings and those of Gregory's edition ; he also noted here and there readings from Vat. 1038 and various Paris MSS. He there- fore pointed the way towards a better text, but committed the error of correcting the Basel text instead of rejecting it altogether and starting afresh. 1824-1825. A most valuable edition of Books I. — VI. is that of J. G. Camerer (and C. F, Hauber) in two volumes published at Berlin*. The Greek text is based on Peyrard, although the Basel and Oxford editions were also used. There is a Latin translation and a collection of notes far more complete than any other I have seen and well nigh inexhaustible. There is no editor or commentator of any mark who is not quoted from ; to show the variety of important authorities drawn upon by Camerer, I need only mention the following names : Proclus, Pappus, Tartaglia, Command inus, Clavius, Peletier, Barrow, Borelli, Watlis, Tacquet, Austin, Simson, Playfair. No words of praise would be too warm for this veritable encyclopaedia of information. 1825. J. G. C. Neide edited, from Peyrard, the text of Books I. — VI., XI. and XII. (Halts Saxoniae). 1826-9. The last edition of the Greek text before Heiberg's is that of E. F. August, who followed the Vatican MS. more closely than Peyrard did, and consulted at all events the Viennese MS. Gr, 103 (Heiberg's V). August's edition (Berlin, 1826-9) contains Books I. — XIII. 1 Eudidis quae supersuni. Las tEuvres a*Eudide t m Gree, en Latin ei en Fronfais d'aprls an msnuscrit tris-atvitn, qui Itait rerte* iwonnu jusqn'a nos jeurt. Par F, Peyrard. Ouvrijje approuv* par I'lnttitut de Franc* (Paris, chei M. Patris). 1 Euclidis elemettforum hbri lex prwrrs graice et Inline tommentarie e scriptis veierum ae rteentforum mathtmatxarum ei Pjltiaertri maxime illustraii [Berolnii, suEnptibus G. Rcimeri). Tom. I. 1814 ; torn. II. t8i{. io4 INTRODUCTION [ch. viii III. Latin versions or commentaries after 1533. 1545. Petrus Ramus (Pierre de la Ramee, 151 5-1572) is credited with a translation of Euclid which appeared in 1545 and again in 1549 at Paris 1 . Ramus, who was more rhetorician and logician than geometer, also published in his Scholae mathematical ( iSS9i Frankfurt; 1569, Base!) what amounts to a series of lectures on Euclid's Elements, in which he criticises Euclid's arrangement of his propositions, the definitions, postulates and axioms, all from the point of view of logic. 1557. Demonstrations to the geometrical Elements of Euclid, six Books, by Peletarius (Jacques Peletier). The second edition (1610) contained the same with the addition of the "Greek text of Euclid"; but only the enunciations of the propositions, as well as the defini- tions etc., are given in Greek (with a Latin translation), the rest is in Latin only. He has some acute observations, for instance about the "angle" of contact •559- Johannes Buteo, or Borrel (1492-1573), published in an appendix to his book De quadratura circuit some notes " on the errors of Campanus, Zambertus, Orontius, Peletarius, Pena, interpreters of Euclid." Buteo in these notes proved, by reasoned argument based on original authorities, that Euclid himself and not Theon was the author of the proofs of the propositions. 1 566. Franciscus Flussates Candalla (Francois de Foix, Comte de Candale, 1 502-1 594) "restored" the fifteen Books, following, as he says, the terminology of Zamberti's translation from the Greek, but drawing, for his proofs, on both Campanus and Theon (i.e. Zamberti) except where mistakes in them made emendation necessary. Other editions followed in 1578, 1602, 1695 (in Dutch). 1572. The most important Latin translation is that of Com- mand irtus (1509-1575) of Urbino, since it was the foundation of most translations which followed it up to the time of Peyrard, including that of Simson and therefore of those editions, numerous in England, which give Euclid "chiefly after the text of Simson." Simson 's first (Latin) edition (1756) has "ex versione Latina Federici Commandini" on the title-page. Commandinus not only followed the original Greek more closely than his predecessors but added to his translation some ancient scholia as well as good notes of his own. The title of his work is Euclidis elementorum libri xv, una cum sckeliis antiquis. A Federico Contmandino Urbinate nuper in latinum conversi, commentariisque quibusdam illustrati (Pisauri, apud Camillum Francischinum). He remarks in his preface that Orontius Finaeus had only edited six Books without reference to any Greek ms., that Peletarius had followed Campanus' version from the Arabic rather than the Greek text, and that Candalla had diverged too far from Euclid, having rejected as inelegant the proofs given in the Greek text and substituted faulty proofs of his own. Commandinus appears to have 1 Described by Boncompagni, BuiUitino, it. p. 3S9. ch. vm] TRANSLATIONS AND EDITIONS 105 used, in addition to the Basel editio princeps, some Greek ms., so far not identified ; he also extracted his " scholia antiqua " from a US. of the class of Vat. 192 containing the scholia distinguished by Heiberg as " Schol. Vat." New editions of Commandinus' translation followed in 1575 (in Italian), 1619, 1749 (in English, by Keill and Stone), 1756 (Books 1. — VI., XI., XII. in Latin and English, by Simson), 1763 (Keill). Besides these there were many editions of parts of the whole work, e.g. the first six Books. [574. The first edition of the Latin version by Clavius' (Christoph Klau [?], born at Bamberg 1537, died 1612) appeared in 1574, and new editions of it in 1589, 1591, 1603, 1607, 1612. It is not a translation, as Clavius himself states in the preface, but it contains a vast amount of notes collected from previous commentators and editors, as well as some good criticisms and elucidations of his own. Among other things, Clavius finally disposed of the error by which Euclid had been identified with Euclid of Megara. He speaks of the differences between Campanus who followed the Arabic tradition and the " commentaries of Theon," by which he appears to mean the Euclidean proofs as handed down by Theon ; he complains of predecessors who have either only given the first six Books, or have rejected the ancient proofs and substituted worse proofs of their own, but makes an exception as regards Commandinus, " a geometer not of the common sort, who has lately restored Euclid, in a Latin translation, to his original brilliancy." Clavius, as already stated, did not give a translation of the Elements but rewrote the proofs, com- pressing them or adding to them, where he thought that he could make them clearer. Altogether his book is a most useful work. 1 62 1. Henry Savile's lectures {Praelectiones tresdecttn in prin- cipium EUtnentorum Etulidis Oxoniae habitae MDC.XX., Oxonii 1621), though they do not extend beyond I. 8, are valuable because they grapple with the difficulties connected with the preliminary matter, the definitions etc., and the tacit assumptions contained in the first propositions, 1654 Andre 1 Tacquet's Elementa geometriae planae et solidae containing apparently the eight geometrical Books arranged for general use in schools. It came out in a large number of editions up to the end of the eighteenth century. 1655, Barrow's Euclidis Eiementorum Libri XV breviter demon- strati is a book of the same kind. In the preface (to the edition of [659) he says that he would not have written it but for the fact that Tacquet gave only eight Books of Euclid. He compressed the work into a very small compass (less than 400 small pages, in the edition of 1659, for the whole of the fifteen Books and the Data) by abbre- viating the proofs and using a large quantity of symbols (which, he says, are generally Oughtred's). There were several editions up to 1732 (those of 1660 and 1732 and one or two others are in English). 1 EudiMs clem/ntor um libri XV. Aictuit xvi. de selidorum regaiarium comparatism. Omnes pefipituis dtmonstratiimibm, accurattsqut nhotiii iltustrati. Auclorc ChristophoTO Ctaaio (Romae, apud Vinccotium Accoltum), 1 vols. to6 INTRODUCTION [ch. vm 1658. Giovanni Alfonso Borelli (1608- 1 679) published Euclides restitutus, on apparently similar lines, which went through three more editions (one in Italian, 1663). 166a Claude Francois Milliet Dechales' eight geometrical Books of Euclid's Elements made easy. Dechales' versions of the Elements had great vogue, appearing in French, Italian and English as well as Latin. Riccardi enumerates over twenty editions. 1733. Saccheri's Euclides ab omni naevo vindicatus sive const us geometrkus quo stabiliuntur prima ipsa geomelriae principia is important for his elaborate attempt to grove the parallel-postulate, forming an important stage in the history of the development of non- Euclidean geometry. 1756. Simson's first edition, in Latin and in English. The Latin title is Euclidis elementorum libri priores sex, item undecimus et duo- decimus, ex versione latina Federici Commandtni; sublatis iis quibus olim libri hi a Tkeone, aliisve, vitiali sunt, et quibusdam Euclidis demonstrationibus restitutis. A Roberto Simson M.D. Glasguae, in aedibus Academicis excudebant Robertus et Andreas Foulis, Academiae typographi. 1802. Euclidis elementorum libri priores XII ex Commandini et Gregorii versianibus latinis. In usum juventutis Academicae..,hy Samuel Horsley, Bishop of Rochester. (Oxford, Clarendon Press.) IV. Italian versions or commentaries. 1543. Tartaglia's version, a second edition of which was pub- lished in 1565', and a third in 1585. It does not appear that he used any Greek text, for in the edition of 1565 he mentions as available only "the first translation by Campano," "the second made by Bartolomeo Zamberto Veneto who is still alive," "the editions of Paris or Germany in which they have included both the aforesaid translations," and "our own translation into the vulgar (tongue)." IS7S- Commandinus' translation turned into Italian and revised by him. 1613. The first six Books "reduced to practice" by Pietro Antonio Cataldi, re-issued in 1620, and followed by Books VH, — ix. (1621) and Book X. (1625). 1663. Borelli's Latin translation turned into Italian by Domenico Magni. 1680. Euclide restitute by Vitale Giordano. 169a Vincenzo Vivian i's Eletnenti piani e solid t di Euclide (Book v. in 1674). 1 The title-page of the edition of 1 56; is ss follows : Euclide Megarcnse phihsophv. sail introduttorc deile identic msthematicc, diligtnttmente rasiettatc, et alia inicgrita ridotls, per it degno pr&fessore di tat identic Nicola Tartatca Briscians, scconde U due tratioltioni. con una ampla espositione delle istcsso tradottore di nuouo aggiutita. ialmente ckiara, cat ogni mediocre ingigno, stitta la notiiia, aver tvffragio di alcutT a/tra icientia con facUita jrrd capacc a peter la inttnderi. In Venetia, Appresso Curtio Troiano, 1565. ch. vm] TRANSLATIONS AND EDITIONS 107 173 1. Elementi geomttrici ptant e solid 'i di Eu elide by Guido Grandi. No translation, but an abbreviated version, of which new editions followed one another up to 1806. 1749. Italian translation of" Dechales with Ozanam's corrections and additions, re-issued 1785, 1797. 1752. Leonardo Ximenes (the first six Books). Fifth edition, 1 819. 181 8. Vincenzo Flauti's Corso di geometries eUtnentare e sublime (4 vols.) contains (Vol. I.) the first six Books, with additions and a dissertation on Postulate 5, and (Vol. II.) Books xi„ XH. Flauti also published the first six Books in 1827 and the Elements of geometry of Euclid m 1843 and 1854 V. German. 1558. The arithmetical Books vir. — ix. by Scheubel" (cf. the edition of the first six Books, with enunciations in Greek and Latin, mentioned above, under date 1 5 50). 1562. The version of the first six Books by Wilhelm Holtzmann (Xylander)*. This work has its interest as the first edition in German, but otherwise it is not of importance. Xylander tells us that it was written for practical people such as artists, goldsmiths, builders etc., and that, as the simple amateur is of course content to know facts, without knowing how to prove them, he has often left out the proofs altogether. He has indeed taken the greatest possible liberties with Euclid, and has not grappled with any of the theoretical difficulties, such as that of the theory of parallels. 1651. Heinrich Hoffmann's Teutscher Euclides (2nd edition 1653), not a translation. 1694. Ant Ernst Burkh. v. Pirckenstein's Teutsch Redender Euclides (eight geometrical Books), "for generals, engineers etc." "proved in a new and quite easy manner." Other editions 1699, 1744. 1697. Samuel Reyher's In teutscher Sfiraclie vorgestellter Euclides (six Books), "made easy, with symbols algebraical or derived from the newest art of solution." 1714. Euclidu xv Bilcker teutsch, "treated in a special and brief manner, yet completely," by Chr. Schessler (another edition in 1729). 1773. The first six Books translated from the Greek for the use of schools by J. F, Lorenz. The first attempt to reproduce Euclid in German word for word. 1 78 1. Books XI., xji. by Lorenz (supplementary to the pre- ceding). Also Euklid's Etemente fUnfzekn Biicker translated from 1 Das libcndochl und taunt buck da kochherumbun Mathematici Euclidis Mtgartnsis... durck Magistrum Jahaun Sckeybf, der loblichen univcrsittt zu Tubingen, des EucHdis und Arithmetic Ordinarien t uuss dem lutein ins teutsch gebra£kt * Di* seeks erste Hucher Euclidis vom an fang oder grund der Geometry. ..Asess Grserhtscker sfiraek in die Teutsch gebraekt aigentlick erkldrt. . . Demassen vcr/uals in Teutscher spraek nie imhen warden... Durch Wilktlm Holttman gtnattt Xylander von Augsfurg. Getnickht m io8 INTRODUCTION [ch. viii the Greek by Lorenz (second edition 1798; editions of 1809, 1818, 1824 by Mollweide, of 1840 by Dippe). The edition of 1824, and I presume those before it, are shortened by the use of symbols and the compression of the enunciation and "setting-out" into one. 1807. Books 1.— vi., xi. ( xil. "newly translated from the Greek," by J. K. F. Hauff. 1828. The same Books by Joh. Jos. Ign. Hoffmann "as guide to instruction in elementary geometry," followed in 1832 by observa- tions on the text by the same editor. 1833. Die Geometric des Euklid und das Wesen derselben by E. S. Unger; also 1838, 1851. 1901. Max Simon, Euclid und die seeks planimetrischen Bucket. VI, French. 1564-1566. Nine Books translated by Pierre Forcadel, a pupil and friend of P. de la Ramee. 1604. The first nine Books translated and annotated by Jean Errard de Bar-Ie-Duc ; second edition, 1605. 161 5. Denis Hen r ion's translation of the 15 Books (seven editions up to 1676), 1639. The first six Books "demonstrated by symbols, by a method very brief and intelligible," by Pierre Hengone, mentioned by Barrow as the only editor who, before him, had used symbols for the exposition of Euclid. 1672. Eight Books "rend us plus faciles" by Claude Francois Mi Diet Dec hales, who also brought out Les ilemens d'Euclide ex- pliquis d'une maniere nouvelie et trh facile, which appeared in many editions, 1672, 1677, 1683 etc. (from 1709 onwards revised by Ozanam), and was translated into Italian (1749 etc.) and English (by William Halifax, 1685). 1 804. In this year, and therefore before his edition of the Greek text, F. Peyrard published the Elements literally translated into French. A second edition appeared in 1 809 with the addition of the fifth Book, As this second edition contains Books 1. — vi. XL, xn. and x. 1, it would appear that the first edition contained Books 1. — iv., VI., XI., XII. Peyrard used for this translation the Oxford Greek text and Simson. VII. Dutch. 1606. Jan Pieterszoon Dou (six Books). There were many later editions. Kastner, in mentioning one of 1702, says that Dou explains in his preface that he used Xylander's translation, but, having after- wards obtained the French translation of the six Books by Errard de Bar-le-Duc (see above),' the proofs in which sometimes pleased him more than those of the German edition, he made his Dutch version by the help of both. 1 61 7. Frans van Schooten, "The Propositions of the Books of Euclid's Elements"; the fifteen Books in this version " enlarged " by Jakob van Leest in 1662. 1695. C. J. Vooght, fifteen Books complete, with Candalla's " 16th." ch. vin] TRANSLATIONS AND EDITIONS 109 1702. Kendfik Coets, six Books (also in Latin, 1692); several editions up to 1752. Apparently not a translation: but an edition for school use. 1763. Pybo Steenstra, Books I. — vr„ XI., XII., likewise an abbre- viated version, several times reissued until 1825. VIII. English. 1570 saw the first and the most important translation, that of Sir Henry Billingsley. The title-page is as follows : THE ELEMENTS OF GEOMETRIE of the most auncient Philosopher E VCLWE of Megara Faithfully (now first) translated into the English toung, by H. Billingsley, Citizen of London. Whereunto are annexed certaine Scholies, Annotations, and Inuentions, of the best Mathematiciens, both of time past, and in this our age. With a very fruitfull Preface by M. I. Dee, specifying the chiefe Mathetnaticall Sciiees, what they are, and whereunto commodious: where, also, are disclosed certaine new Secrets Mathetnaticall and Afechanicall, vntill these our dales, greatly missed. Imprinted at London by John Daye. The Preface by the translator, after a sentence observing that with- out the diligent study of Euclides Efementes it is impossible to attain unto the perfect knowledge of Geometry, proceeds thus. " Wherefore considering the want and lacke of such good authors hitherto in our Englishe tounge, lamenting also the negligence, and lacke of zeale to their countrey in those of our nation, to whom God hath geuen both knowledge and also abilitie to translate into our tounge, and to publishe abroad such good authors and bookes (the chiefe instrumentes of all learninges): seing moreouer that many good wittes both of gentlemen and of others of all degrees, much desirous and studious of these artes, and seeking for them as much as they can, sparing no paines, and yet frustrate of their intent, by no meanes attaining to that which they seeke : I haue for their sakes, with some charge and great trauaile, faithfully translated into our vulgar e touge, and set abroad in Print, this booke of Euclide. Whereunto I haue added easie and plaine declarations and examples by figures, of the defini- tions. In which booke also ye shall in due place finde manifolde additions, Scholies, Annotations, and Inuentions; which I haue gathered out of many of the most famous and chiefe Mathematicies, both of old time, and in our age : as by diligent reading it in course, ye shall well perceaue...," It is truly a monumental work, consisting of 464 leaves, and there- fore 928 pages, of folio size, excluding the lengthy preface by Dee. The notes certainly include all the most important that had ever been tto INTRODUCTION [cm. vin written, from those of the Greek commentators, Proclus and the others whom he quotes, down to those of Dee himself on the la3t books. Besides the fifteen Books, Billingsley included the "sixteenth" added by Candalla, The print and appearance of the book are worthy of its contents ; and, in order that it may be understood how no pains were spared to represent everything in the clearest and most perfect form, I need only mention that the figures of the propositions in Book XI. are nearly all duplicated, one being the figure of Euclid, the other an arrangement of pieces of paper (triangular, rectangular etc.) pasted at the edges on to the page of the book so that the pieces can be turned up and made to show the real form of the solid figures represented. Billingsley was admitted Lady Margaret Scholar of St John's College, Cambridge, in 1551, and he is also said to have studied at Oxford, but he did not take a degree at either University. He was afterwards apprenticed to a London haberdasher and rapidly became a wealthy merchant. Sheriff of London in 1584, he was elected Lord Mayor on 31st December, 1596, on the death, during his year of office, of Sir Thomas Skinner. From 1589 he was one of the Queen's four " customers," or farmers of customs, of the port of London. In 1 591 he founded three scholarships at St John's College for poor students, and gave to the College for their maintenance two messuages and tenements in Tower Street and in Mark Lane, Allhallows, Barking. He died in 1606. 1651. Elements of Geometry. The first VI Bootks: In a compen- dious form contracted and demonstrated by Captain Thomas Rudd, with the mathematicall preface of John Dee (London). 1660. The first English edition of Barrow's Euclid (published in Latin in 1655)1 appeared in London. It contained "the whole fifteen books compendiously demonstrated"; several editions followed, in 1705, 1722, 1732, 1751. 1 66 1. Euclid 's Elements of Geometry, with a supplement of divers Propositions and Corollaries. To which is added a Treatise of regular Soli lis by Campane and Ftussat ; likewise Euclid's Data and Marinus his Preface. Also a Treatise of the Divisions of Superficies, ascribed to Machomet Bagdedine, but published by Commandine at the request of J, Dee of London, Published by care and industry of John Leeke and Geo. Serle, students in the Math. (London). According to Potts this was a second edition of Billingsley's translation. 1685. William Halifax's version of Dechales' " Elements of Euclid explained in a new but most easy method " (London and Oxford). 1705. The English Euclide; being tlie first six Elements of Geometry, translated out of the Greek, with annotations and useful/ supplements by Edmund Scarburgh (Ox ford ). A noteworthy and useful edition. 1708. Books I. — VL, XL, xii., translated from Command in us' Latin version by Dr John Keill, Savilian Professor of Astronomy at Oxford. Keill complains in his preface of the omissions by such editors as Tacquet and Dechales of many necessary propositions (e.g. VI. 27 — 29), and of their substitution of proofs of their own for Euclid's. He praises Barrow's version on the whole, though objecting to the " algebraical " ca. viii] TRANSLATIONS AND EDITIONS in form of proof adopted in Book n., and to the excessive use of notes and symbols, which (he considers) make the proofs too short and thereby obscure: his edition was therefore intended to hit a proper mean between Barrow's excessive brevity and Clavius' prolixity. Keill's translation was revised by Samuel Cunn and several times reissued. 1749 saw the eighth edition, 1772 the eleventh, and 1782 the twelfth. 1714. W. Whiston's English version (abridged) of The Elements of Euclid with select theorems out of Archimedes by the learned Andr. Tacquet. 1756. Simson's first English edition appeared in the same year as his Latin version under the title : The Elements of Euclid, vis. the first six Books together with the eleventh and twelfth. In this Edition the Errors by which Theon or others have long ago vitiated these Books are corrected and some of Euclid's Demonstrations are restored. By Robert Simson (Glasgow). As above stated, the Latin edition, by its title, purports to be " ex version© latina Federici Commandini," but to the Latin edition, as well as to the English editions, are appended Notes Critical and Geometrical ; containing an Account of those things in which this Edition differs from the Greek text; and the Reasons of the Alterations which have been made. As also Obser- vations on some of the Propositions. Simson says in the Preface to some editions (e.g. the tenth, of 1799) that "the translation is much amended by the friendly assistance of a learned gentleman." Simson's version and his notes are so well known as not to need any further description. The book went through some thirty suc- cessive editions. The first five appear to have been dated 1756, 1762, 1767, 1772 and 1775 respectively; the tenth 1799, the thirteenth 1806, the twenty-third 1830, the twenty-fourth 1834, the twenty-sixth 1844. The Data "in like manner corrected '' was added for the first time in the edition of 1762 (the first octavo edition). 1781, 1788. In these years respectively appeared the two volumes containing the complete translation of the whole thirteen Books by James Williamson, the last English translation which reproduced Euclid word for word. The title is The Elements of Euclid, with Dissertations intended to assist and encourage a critical examination of these Elements, as the most effectual means of establishing a j'uster taste upon mathematical subjects than that which at present prevails. By James Williamson. In the first volume (Oxford, 1781) he is described as " M.A. Fellow of Hertford College," and in the second (London, printed by T. Spilsbury, 1788) as "B.D." simply. Books v., VI. with the Con- clusion in the first volume are paged separately from the rest 1 78 1 . 4 n examination of the first six Books of Euclid's Elements, by William Austin (London). '79S- John Playfair's first edition, containing "the first six Books of Euclid with two Books on the Geometry of Solids." The book ii3 INTRODUCTION [ch. vm reached a fifth edition in 1819, an eighth in 1831, a ninth in 1836, and a tenth in 1846. 1826. Riccardi notes under this date Euclid's Elements of Geo- metry containing the whole twelve Books translated into English, from the edition of Peyrard, by George Phillips. The editor, who was President of Queens' College, Cambridge, 1857-1892, was born in 1804 and matriculated at Queens' in 1 826, so that he must have published the book as an undergraduate. 1828. A very valuable edition of the first six Books is that of Dionysius Lardner, with commentary and geometrical exercises, to which he added, in place of Books XI., XII., a Treatise on Solid Geometry mostly based on Legend re, Lardner compresses the pro- positions by combining the enunciation and the setting-out, and he gives a vast number of riders and additional propositions in smaller print The book had reached a ninth edition by 1846, and an eleventh by 1855. Among other things, Lardner gives an Appendix "on the theory of parallel lines," in which he gives a short history of the attempts to get over the difficulty of the parallel' postulate, down to that of Legendre. 1833. T. Perronet Thompson's Geometry without axioms, or the first Book of Euclid's Elements with alterations and notes ; and an intercalary book in which the straight line and plane are derived from properties of the sphere, with an appendix containing notices of methods proposed for getting over the difficulty in the twelfth axiom of Euclid. Thompson (1783-1869) was 7th wrangler 1802, midshipman 1803, Fellow of Queens* College, Cambridge, 1804, and afterwards general and politician. The book went through several editions, but, having been well translated into French by Van Tenac, is said to have received more recognition in France than at home. 1 845. Robert Potts' first edition (and one of the best) entitled : Euclid's Elements of Geometry chiefly from the text of Dr Simson with explanatory notes... to which is prefixed an introduction containing a brief outline of the History of Geometry. Designed for the use of the higher forms in Public Schools and students in the Universities (Cambridge University Press, and London, John W. Parker), to which was added (1847) An Appendix to the larger edition of Euclid's Elements of Geometry, containing additional notes on the Elements, a short tract on trans- versals, and hints for the solution of the problems etc. 1862. Todhunter's edition. The later English editions 1 will not attempt to enumerate; their name is legion and their object mostly that of adapting Euclid for school use, with all possible gradations of departure from his text and order. IX, Spanish. 1576. The first six Books translated into Spanish by Rodrigo Camorano. 1637. The first six Books translated, with notes, by L. Carduchi. 1689. Books 1.— vi, XI, XII, translated and explained by Jacob Knesa. ch. vni] TRANSLATIONS AND EDITIONS 113 X. Russian. 1739. Ivan Astaroff* (translation from Latin). 1 789. Pr. Suvoroff and Yos. Nikitin {translation from Greek). 1880. Vachtchenko-Zakhartchenko. {181 7, A translation into Polish by Jo. Czecha.) XI. Swedish. 1744. Mitten Stromer, the first six Books ; second edition 1748. The third edition (1753) contained Books XI.— XU. as well; new editions continued to appear till 1884, 1836. H. Falk, the first six Books. 1844, 1845, 1859. P. R. Brakenhjelm, Books I. — VI., XL, xn. 1850. F. A. A. Lundgren. 1850. H. A. Witt and M. E. Areskong, Books I.— VI., XI., XII. XII. Danish. 1745. Ernest Gottlieb Ziegenbalg. 1803. H. C. Linderup, Books I. — VI. XIII. Modern Greek. 1820. Benjamin of Lesbos, I should add a reference to certain editions which have appeared in recent years. A Danish translation (Euklid's Eletnenter oversat af Thyra Eibe) was completed in 1912 ; Books I. — II. were published (with an Intro- duction by Zeuthen) in 1897, Books in. — IV. in 1900, Books v. — VI. in 1904, Books VII. — XIII. in 19 1 2, The Italians, whose great services to elementary geometry are more than once emphasised in this work, have lately shown a note- worthy disposition to make the ipsissima verba of Euclid once more the object of study. Giovanni Vacca has edited the text of Book I. (// prima libro degli Elementi. Testo greco, versione italiana, intro- duzione e note, Firenze 1 916,) Federigo Enriques has begun the publication of a complete Italian translation (Gli Elementi d' Enclide e la critica antica e moderna); Books I. — IV. appeared in 1925 (Alberto Stock, Roma). An edition of Book I. by the present writer was published in 1918 {Euclid in Greek \ Book [., with Introduction and Notes, Camb. Univ. Press). CHAPTER IX. j I, ON THE NATURE OF ELEMENTS. It would not be easy to find a more lucid explanation of the terms element and elementary, and of the distinction between them, than is found in Prod us \ who is doubtless, here as so often, quoting from Geminus. There are, says Proclus, in the whole of geometry certain leading theorems, bearing to those which follow the relation of a principle, all- pervading, and furnishing proofs of many properties. Such theorems are called by the name of elements ; and their function may be compared to that of the letters of the alphabet in relation to language, letters being indeed called by the same name in Greek (orotyeta). The term elementary, on the other hand, has a wider application : it is applicable to things " which extend to greater multiplicity, and, though possessing simplicity and elegance, have no longer the same dignity as the elements, because their investigation is not of general use in the whole of the science, e.g. the proposition that in triangles the perpendiculars from the angles to the transverse sides meet in a point." " Again, the term element is used in two senses, as Menaechmus says. For that which is the means of obtaining is an element of that which is obtained, as the first proposition in Euclid is of the second, and the fourth of the fifth. In this sense many things may even be said to be elements of each other, for they are obtained from one another. Thus from the fact that the exterior angles of rectilineal figures are (together) equal to four right angles we deduce the number of right angles equal to the internal angles (taken together)*, and vice versa. Such an element is like a lemma. But the term element is otherwise used of that into which, being more simple, the composite is divided ; and in this sense we can no longer say that everything is an element of everything, but only that things which are more of the nature of principles are elements of those which stand to them in the relation of results, as postulates are elements of theorems. It is 1 Proclus, Coium, on Eucl. I., ed. Friedlein, pp. Jssqq. * t4 TrkTjSin rw irrit dfi0a(j tnur. If the te*t is right, we must apparently take it as "the number of the angles equal to right angles that there are inside/' i.e. that are made up by the internal angles. ch. ix. J i] ON THE NATURE OF ELEMENTS 115 according to this signification of the term element that the elements found in Euclid were compiled, being partly those of plane geometry, and partly those of stereometry. In like manner many writers have drawn up elementary treatises in arithmetic and astronomy. " Now it is difficult, in each science, both to select and arrange in due order the elements from which all the rest proceeds, and into which all the rest is resolved. And of those who have made the attempt some were able to put together more and some less ; some used shorter proofs, some extended their investigation to an indefinite length ; some avoided the method of reductio ad absurdum, some avoided proportion-, some contrived preliminary steps directed against those who reject the principles ; and, in a word, many different methods have been invented by various writers of elements. "It is essential that such a treatise should be rid of everything superfluous (for this is an obstacle to the acquisition of knowledge) ; it should select everything that embraces the subject and brings it to a point (for this is of supreme service to science) ; it must have great regard at once to clearness and conciseness (for their opposites trouble our understanding); it must aim at the embracing of theorems in general terms (for the piecemeal division of instruction into the more partial makes knowledge difficult to grasp). In all these ways Euclid's system of elements will be found to be superior to the rest ; for its utility avails towards the investigation of the primordial figures 1 , its clearness and organic perfection are secured by the progression from the more simple to the more complex and by the foundation of the investigation upon common notions, while generality of demonstration is secured by the progression through the theorems which are primary and of the nature of principles to the things sought. As for the things which seem to be wanting, they are partly to be discovered by the same methods, like the construction of the scalene and isosceles (triangle), partly alien to the character of a selection of elements as introducing hopeless and boundless complexity, like the subject of unordered irrationals which Apoilonius worked out at length", and partly developed from things handed down (in the elements) as causes, like the many species of angles and of lines. These things then have been omitted in Euclid, though they have received full discussion in other works ; but the knowledge of them is derived from the simple (elements)." Proclus, speaking apparently on his own behalf, in another place distinguishes two objects aimed at in Euclid's Elements. The first has reference to the matter of the investigation, and here, like a good Platonist, he takes the whole subject of geometry to be concerned with the "cosmic figures," the five regular solids, which in Book XHi. 1 rwt Apxix&r <jyv!i&Twi\ by which Procius probably means the regular polyhedra (Tannery, p. l43ff.). ■ We have no more than the most obscure indications of the character of this work in an Arabic MS. analysed by Wocpcke, Essai d*une rtstiiutitm dt travaux pirdus tfApoUojnus sur Us quantities irraiwntlUs d'aprls des indications tirhs tTun manuscrit arabe in Mtmvirtx prlttnttt e fatadMit da Mentis, xjv. 658—710, Paris, 1856. Cf. Cantor, Gtsth. d. Math. t 3 , pp. 34& — 9: details are also given in my notes to Book X. n6 INTRODUCTION [ch ex. § i are constructed, inscribed in a sphere and compared with one another. The second object is relative to the learner; and, from this standpoint, the elements may be described as "a means of perfecting the learner's understanding with reference to the whole of geometry. For, starting from these (elements), we shall be able to acquire knowledge of the other parts of this science as well, while without them it is impossible for us to get a grasp of so complex a subject, and knowledge of the rest is unattainable. As it is, the theorems which are most of the nature of principles, most simple, and most akin to the first hypotheses are here collected, in their appropriate order ; and the proofs of all other propositions use these theorems as thoroughly well known, and start from them. Thus Archimedes in the books on the sphere and cylinder, Apollonius, and all other geometers, clearly use the theorems proved in this very treatise as constituting admitted principles 1 ." Aristotle too speaks of elements of geometry in the same sense. Thus: "in geometry it is well to be thoroughly versed in the elements* " ; " in general the first of the elements are, given the definitions, e.g. of a straight line and of a circle, most easy to prove, although of course there are not many data that can be used to establish each of them because there are not many middle terms'"; "among geometrical propositions we call those 'elements' the proofs of which are contained in the proofs of all or most of such propositions'"; "(as in the case of bodies), so in like manner we speak of the elements of geometrical propositions and, generally, of demonstrations ; for the demonstrations which come first and are contained in a variety of other demonstrations are called elements of those demonstrations... the term element is applied by analogy to that which, being one and small, is useful for many purposes •." § x ELEMENTS ANTERIOR TO EUCLID'S. The early part of the famous summary of P rod us was no doubt drawn, at least indirectly, from the history of geometry by Eudemus ; this is generally inferred from the remark, made just after the mention of PhiHppus of Medma, a disciple of Plato, that "those who have written histories bring the development of this science up to this point." We have therefore the best authority for the list of writers of elements given in the summary. Hippocrates of Chios (fl, in second half of 5th c.) is the first; then Leon, who also discovered diorismi, put together a more careful collection, the propositions proved in it being more numerous as well as more serviceable*. Leon was a little older than Eudoxus (about 408-355 n.C.) and a little younger than Plato (428/7-347/6 B.C.), but Hid not belong to the latter's school. The J Proclus, pp. 70, 10— 71, 31. 5 Tapks vm. 14,163b 13.' % Topics vm. 3, 158 b 35. ' Mttaph. 908 » )j, ' Mttaph. 1014 a 35 — b 5, Proclus, p. 66, 10 UMTre rbv Alojrra xal rd ffT«x*ia rvpfftiftu t^j re w\f)8ti teal r£ xpfi? ch. ix. (»] ELEMENTS ANTERIOR TO EUCLID'S 117 geometrical text-book of the Academy was written by Theudius of Magnesia, who, with Amyclas of Heraclea, Menaechmus the pupil of Eudoxus, Menaechmus' brother Dinostratus and Athenaeus of Cyzicus consorted together in the Academy and carried on their investigations in common. Theudius " put together the elements admirably, making many partial (or limited) propositions more general 1 ." Eudemus mentions no text- book after that of Theudius, only adding that Her- motimus of Colophon "discovered many of the elements 1 ." Theudius then must be taken to be the immediate precursor of Euclid, and no doubt Euclid made full use of Theudius as well as of the discoveries of Hermotimus and all other available material. Naturally it is not in Euclid's Elements that we can find much light upon the state of the subject when he took it up ; but we have another source of informa- tion in Aristotle. Fortunately for the historian of mathematics, Aristotle was fond of mathematical illustrations ; he refers to a con- siderable number of geometrical propositions, definitions etc., in a way which shows that his pupils must have had at hand some text- book where they could find the things he mentions; and this text -book must have been that of Theudius. Heiberg has made a most valuable collection of mathematical extracts from Aristotle*, from which much is to be gathered as to the changes which Euclid made in the methods of his predecessors ; and these passages, as well as others not included in Heiberg's selection, will often be referred to in the sequel. §3. FIRST PRINCIPLES: DEFINITIONS, POSTULATES, AND AXIOMS. On no part of the subject does Aristotle give more valuable information than on that of the first principles as, doubtless, generally accepted at the time when he wrote. One long passage in the Posterior Analytics is particularly full and lucid, and is worth quoting m txtenso. After laying it down that every demonstrative science starts from necessary principles', he proceeds': " By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first (terms) and those derived from them is assumed ; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed); and the existence of the unit and of magnitude must also be assumed, but the rest must be proved. Now of tie premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are applied to the sub- ject-matter included under the particular science. Instances of first 1 Proclus, p. 67, 14 «oi yip ri jtoix'" 1 *"\ut avriragtr ml ro\Wl rur ntpiKwi [Apiieup (?) Fried lei n ] Ka8o\uttitTtpa (watipfy. 1 PrOClliS, p. 67, 11 TWf tiraixtluv roWa. ivtvp*. 1 Matktmatutha i» Arisleitlts in Ad/wndtungen fur Gach. d. math. Wisstnschafttn, XVIII. Heft (1904), pp. 1—49. * Anal. pest. I. 6, 74 b 5. ' **»<£ I. 'O, 76 a 31 — 77 a 4. n8 INTRODUCTION [ch. ix. j 3 principles peculiar to a science are the assumptions that a line is of such and such a character, and similarly for the straight (line); whereas it is a common principle, for instance, that, if equals be subtracted from equals, the remainders are equal. But it is enough that each of the common principles is true so far as regards the particular genus (subject-matter) ; for (in geometry) the effect will be the same even if the common principle be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers. " Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and l : nes. With these things it is assumed that they exist and that they are of such and such a nature. But, with regard to their essential properties, what is assumed is only the meaning of each term employed : thus arithmetic assumes the answer to the question what is (meant by) ' odd ' or 'even,' 'a square' or 'a cube,' and geometry to the question what is (meant by) ' the irrational ' or ' deflection ' or (the so-called) ' verging ' (to a point) ; but that there are such things is proved by means of the common principles and of what has already been demonstrated. Similarly with astronomy. For every demonstrative science has to do with three things, (i) the things which are assumed to exist, namely the genus (subject-matter) in each case, the essential properties of which the science investigates, (2) the common axioms so-called, which are the primary source of demonstration, and (3) the properties with regard to which all that is assumed is the meaning of the respective terms used. There is, however, no reason why some sciences should not omit to speak of one or other of these things. Thus there need not be any supposition as to the existence of the genus, if it is manifest that it exists (for it is not equally clear that number exists and that cold and hot exist) ; and, with regard to the properties, there need be no assumption as to the meaning of terms if it is clear r just as in the common (axioms) there is no assumption as to what is the meaning of subtracting equals from equals, because it is well known. But none the less is it true that there are three things naturally distinct, the subject-matter of the proof, the things proved, and the (axioms) from which (the proof starts). "Now that which isperse necessarily true, and must necessarily be thought so, is not a hypothesis nor yet a postulate. For demon- stration has not to do with reasoning from outside but with the reason dwelling in the soul, j ust as is the case with the syllogism. It is always possible to raise objection to reasoning from outside, but to contradict the reason within us is not always possible. Now anything that the teacher assumes, though it is matter of proof, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis, not abso- lutely, but relatively to the particular pupil ; but, if the same thing is assumed when the learner either has no opinion on the subject or is of a contrary opinion, it is a postulate. This is the difference ch. ix. §3] FIRST PRINCIPLES 119 between 3 hypothesis and a postulate ; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or whatever is assumed and used without being proved, although matter for demonstration. Now definitions are not hypotheses, for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood : a definition is therefore not a hypothesis, unless indeed it be asserted that any audible speech is a hypothesis. A hypothesis is that from the truth of which, if assumed, a conclusion can be established. Nor are the geometer's hypotheses false, as some have said : I mean those who say that ' you should not make use of what is false, and yet the geometer falsely calls the line which he has drawn a foot long when it is not, or straight when it is not straight' The geometer bases no conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is illustrated by the figures. Further, the postulate and every hypothesis are either universal or particular statements; definitions are neither" (because the subject is of equal extent with what is predicated of it). Every demonstrative science, says Aristotle, must start from in- demonstrable principles : otherwise, the steps of demonstration would be endless. Of these indemonstrable principles some are (a) common to all sciences, others are (0) particular, or peculiar to the particular science; (a) the common principles are the axioms, most commonly illustrated by the axiom that, if equals be subtracted from equals, the remainders are equal. Coming now to (6) the principles peculiar to the particular science which must be assumed, we have first the genus or subject-matter, the existence of which must be assumed, viz. magni- tude in the case of geometry, the unit in the case of arithmetic. Under this we must assume definitions of manifestations or attributes of the genus, e.g. straight lines, triangles, deflection etc. The definition in itself says nothing as to the existence of the thing defined : it only requires to be understood. But in geometry, in addition to the genus and the definitions, we have to assume the existence of a few primary things which are defined, viz. points and lines only : the existence of everything else, e.g. the various figures made up of these, as triangles, squares, tangents, and their properties, e.g, incommensur- ability etc., has to be proved (as it is proved by construction arid demonstration). In arithmetic we assume the existence of the unit: but, as regards the rest, only the definitions, e.g, those of odd, even, square, cube, are assumed, and existence has to be proved. We have then clearly distinguished, among the indemonstrable principles, axioms and definitions, A postulate is also distinguished from a hypothesis, the latter being made with the assent of the learner, the former without such assent or even in opposition to his opinion (though, strangely enough, immediately after saying this, Aristotle gives a wider meaning to "postulate" which would cover "hypothesis" as well, namely whatever is assumed, though it is matter for proof, and used without being proved). Heiberg remarks that there is no trace in Aristotle of Euclid's Postulates, and that " postulate" in Aristotle has iso INTRODUCTION [ch. ix. §3 a different meaning. He seems to base this on the alternative description of postulate, indistinguishable from a hypothesis ; but, if we take the other description in which it is distinguished from a hypothesis as being an assumption of something which is a proper subject of demonstration without the assent or against the opinion of the learner, it seems to fit Euclid's Postulates fairly well, not only the fi rst th ree ( post u I at i ng t h re e con struc t i on s), bu t e m i n en tl y al so the other two, that all right angles are equal, and that two straight lines meeting a third and making the internal angles on the same side of it less than two right angles will meet on that side. Aristotle's description also seems to me to suit the "postulates" with which Archimedes begins his book On the equilibrium of planes, namely that equal weights balance at equal distances, and that equal weights at unequal distances do not balance but that the weight at the longer distance will prevail. Aristotle's distinction also between hypothesis and definition, and between hypothesis and axiom, is clear from the following passage : "Among immediate syllogistic principles, I call that a thesis which- it is neither possible to prove nor essential for any one to hold who is to learn anything ; but that which it is necessary for any one to hold who is to learn anything whatever is an axiom : for there are some principles of this kind, and that is the most usual name by which we speak of them. But, of theses, one kind is that which assumes one or other side of a predication, as, for instance, that something exists or does not exist, and this is a hypothesis ; the other, which makes no Such assumption, is a definition. For a definition is a thesis: thus the arithmetician posits (rlOcTat) that a unit is that which is indivisible in respect of quantity ; but this is not a hypo- thesis, since what is meant by a unit and the fact that a unit exists are different things 1 ." Aristotle uses as an alternative term for axioms "common (things)," to, Kowd, or "common opinions" (koiwiI Sofoi)> as in the following passages. " That, when equals are taken from equals, the remainders are equal is (a) common (principle) in the case of alt quantities, but mathematics takes a separate department (atroXa^ovo'a) and directs its investigation to some portion of its proper subject-matter, as e.g. lines or angles, numbers, or any of the other quantities*.' 1 "The common (principles), e.g. that one of two contradictories must be true, that equals taken from equals etc., and the like" " " With regard to the principles of demonstration, it is questionable whether they belong to one science or to several. By principles of demonstration I mean the common opinions from which all demonstration proceeds, e.g. that one of two contradictories must be true, and that it is impossible for the same thing to be and not be 4 ." Similarly "every demonstrative (science) investigates, with regard to some subject-matter, the essential attributes, starting from the common opinions'." We have then here, as Heiberg says, a sufficient explanation of Euclid's term for axioms, 1 Anal. pest. I. a, 71a 14—14. ' Mttaph. 1061 b 19—14. * Anal. post. \. n, 77 a 30. * Mttaph. 996 b 36 — 30. 5 Mttaph. 907 a so — 11. ch. ix. §3] FIRST PRINCIPLES lai viz. common notions (koivoI evvoiat), and there is no reason to suppose it to be a substitution for the original term due to the Stoics : cf. Proclus' remark that, according to Aristotle and the geometers, axiom and common notion are the same thing 1 . Aristotle discusses the indemonstrable character of the axioms in the Metaphysics. Since "all the demonstrative sciences use the axioms'," the question arises, to what science does their discussion belong*? The answer is that, like that of Being (oiJct-mi), it is the province of the (first) philosopher 1 . It is impossible that there should be demonstration of everything, as there would be an infinite series of demonstrations: if the axioms were the subject of a demonstrative science, there would have to be here too, as in other demonstrative sciences, a subject-genus, its attributes and corresponding axioms 1 \ thus there would be axioms behind axioms, and so on continually. The axiom is the most firmly established of all principles*. It is ignorance alone that could lead anyonetotryto prove the axioms' ; the supposed proof would be a.petilio principii'. If it is admitted that not every- thing can be proved, no one can point to any principle more truly indemonstrable'. If any one thought he could prove them, he could at once be refuted ; if he did not attempt to say anything, it would be ridiculous to argue with him i he would be no better than a vegetable 10 . The first condition of the possibility of any argument whatever is that words should signify something both to the speaker and to the hearer: without this there can be no reasoning with any one. And, if any one admits that words can mean anything to both hearer and speaker, he admits that something can be true without demon- stration. And so on". It was necessary to give some sketch of Aristotle's view of the first principles, if only in connexion with Proclus' account, which is as follows. As in the case of other sciences, so " the compiler of elements in geometry must give separately the principles of the science, and after that the conclusions from those principles, not giving any account of the principles but only of their consequences. No science proves its own principles, or even discourses about them : they are treated as self-evident. . .Thus the first essential was to dis- tinguish the principles from their consequences. Euclid carries out this plan practically in every book and, as a preliminary to the whole enquiry, sets out the common principles of this science. Then he divides the common principles themselves into hypotheses, postulates, and axioms. For all these are different from one another : an axiom, a postulate and a hypothesis are not the same thing, as the inspired Aristotle somewhere says. But, whenever that which is assumed and ranked as a principle is both known to the learner and convincing in itself, such a thing is an axiom, e.g. the statement that things which are equal to the same thing are also equal to one another. When, on 1 Proclus, p. 194, 8. ' Mtlaph. 997 a 10. ' ibid. 996 b 16, * ibid. 100s a 11 — b 11. ' Hid. 997 a 5 — 8. ■ ibid. 1005 b II — 17. ' ibid, loofia 5. s ibid. 1006 a 17. ibid. 1006a 10. I0 ibid. 1006a 11 — 15. 1} ibid. 1006a iSsqq. u» INTRODUCTION [cH.1x.j3 the other hand, the pupil has not the notion of what is told him which carries conviction in itself, but nevertheless lays it down and assents to its being assumed, such an assumption is a hypothesis. Thus we do not preconceive by virtue of a common notion, and without being taught, that the circle is such and such a figure, but, when we are told so, we assent without demonstration. When again what is asserted is both unknown and assumed even without the assent of the learner, then, he says, we call this a postulate, e.g. that all right angles are equal. This view of a postulate is clearly implied by those who have made a special and systematic attempt to show, with regard to one of the postulates, that it cannot be assented to by any one straight off. According then to the teaching of Aristotle, an axiom, a postulate and a hypothesis are thus distinguished 1 ." We observe, first, that Proclus in this passage confuses hypotheses and definitions, although Aristotle had made the distinction quite plain. The confusion may be due to his having in his mind a passage of Plato from which he evidently got the phrase about " not giving an account of" the principles. The passage is a : " I think you know that those who treat of geometries and calculations (arithmetic) and such things take for granted (inroQefievoi) odd and even, figures, angles of three kinds, and other things akin to these in each subject, implying that they know these things, and, though using them as hypotheses, do not even condescend to give any account of them either to themselves or to others, but begin from these things and then go through everything else in order, arriving ultimately, by recognised methods, at the conclusion which they started in search of." But the hypothesis is here the assumption, e.g, ' that there may he sttch a thing as length without breadth, henceforward called a line',' and so on, without any attempt to show that there is such a thing ; it is mentioned in connexion with the distinction between Plato's 'superior' and 'inferior' intellectual method, the former of which uses successive hypotheses as stepping-stones by which it mounts upwards to the idea of Good. We pass now to Proclus' account of the difference between postu- lates and axioms. He begins with the view of Geminus, according to which " they differ from one another in the same way as theorems are also distinguished from problems. For, as in theorems we propose to see and determine what follows on the premisses, while in problems we are told to find and do something, in like manner in the axioms such things are assumed as are manifest of themselves and easily apprehended by our untaught notions, while in the postulates we assume such things as are easy to find and effect (our understanding suffering no strain in their assumption), and we require no complication of machinery*."..." Both must have the characteristic of being simple 1 Proclos, pp. 7S, 10—77, *• ' Republic, VI, 510 c. Cf. Aristotle, jVk. Eth. 1151a 17. * H. Jackson, Journal of Philology, vol. x. p. 144- * Proclus, pp. 178, 1? — (79,8* In illustration Proclus contrasts the drawing of a straight Line or a circle with the drawing of a " single- turn spiral " or of an equilateral triangle, the ch. ix. §3] FIRST PRINCIPLES 123 and readily grasped, I mean both the postulate and the axiom ; but the postulate bids us contrive and find some subject-matter (uXj,) to exhibit a property simple and easily grasped, while the axiom bids us assert some essential attribute which is self-evident to the learner, just as is the fact that fire is hot, or any of the most obvious things 1 ," Again, says Proclus, " some claim that alt these things are alike postulates, in the same way as some maintain that all things that are sought are problems. For Archimedes begins his first book on /«- equilibrium' 1 with the remark ' I postulate that equal weights at equal distances are in equilibrium,' though one would rather call this an axiom. Others call them all axioms in the same way as some regard as theorems everything that requires demonstration'." " Others again will say that postulates are peculiar to geometrical subject-matter, while axioms are common to all investigation which is concerned with quantity and magnitude. Thus it is the geometer who knows that all right angles are equal and how to produce in a straight line any limited straight line, whereas it is a Common notion that things which are equal to the same thing are also equal to one another, and it is employed by the arithmetician and any scientific person who adapts the general statement to his own subject 1 ." The third view of the distinction between a postulate and an axiom is that of Aristotle above described'. The difficulties in the way of reconciling Euclid's classification of postulates and axioms with any one of the three alternative views are next dwelt upon. If we accept the first view according to which an axiom has reference to something known, and a postulate to something done, then the 4th postulate (that all right angles are equal) \% not a postulate ; neither is the 5th which states that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which are the angles less than two right angles. On the second view, the assumption that two straight lines cannot enclose a space, " which even now," says Proclus, " some add as an axiom," and which is peculiar to the subject-matter of geometry, like the fact that all right angles are equal, is not an axiom. According to the third (Aristotelian) view, "everything which is confirmed (irt<not>Tat) by a sort of demonstration spiral requiring more complex machinery and even the equilateral triangle needing a certain method. " For the geometrical intelligence will say that hy conceiving a straight tine fined at one end but, as regards the other end, moving round the fixed end, and a point moving along the straight line from the fined end, 1 have described the single- turn spiral ; for the end of the straight line descrihing a circle, and the point moving on the straight line simul- taneously, when they arrive and meet at the same point, complete such a spiral. And again, if I draw equal circles, join their common point to the centres of the circles and draw a straight line from one of the centres to the other, I shall have the equilateral triangle. These things then are far from being completed by means of a single act or of a moment's thought" (p. 180. S— ji). 1 Proclus, p. i Si, 4 — 11. 1 It is necessary to coin a word to render tontapfxirtur, which is moreover in the plural. The title of the treatise as we have it is Equilibria of plana or centres of gravity of planes in Book r and Equilibria of planes in Book It. • Proclus, p. 181, 16—13. * **S p. 181, 6—14. ■ Pp. 118, 119. i»4 INTRODUCTION [ch. ix. §3 will be a postulate, and what is incapable of proof will be an axiom 1 ." This last statement of Proclus is loose, as regards the axiom, because it omits Aristotle's requirement that the axiom should be a self- evident truth, and one that must be admitted by any one who is to learn anything at all, and, as regards the postulate, because Aristotle calls a postulate something assumed without proof though it is "matter of demonstration" (aTroSetierhv Sv), but says nothing of a quasi -demon strati on of the postulates. On the whole I think it is from Aristotle that we get the best idea of what Euclid understood by a postulate and an axiom or common notion. Thus Aristotle's account of an axiom as a principle common to all sciences, which is self-evident, though incapable of proof, agrees sufficiently with the contents of Euclid's common notions as reduced to five in the most recent text (not omitting the fourth, that " things which coincide are equal to one another"). As regards the postulates, it must be borne in mind that Aristotle says elsewhere' that, "other things being equal, that proof is the better which proceeds from the fewer postulates or hypotheses or propositions." I f then we say that a geometer must lay down as principles, first certain axioms or common notions, and then an irreducible minimum of postulates in the Aristotelian sense concerned only with the subject-matter of geometry, we are not far from describing what Euclid in fact does. As regards the postulates we may imagine him saying : ■ Besides the common notions there are a few other things which I must assume without proof, but which differ from the common notions in that they are not self-evident. The learner may or may not be disposed to agree to them ; but he must accept them at the outset on the superior authority of his teacher, and must be left to convince himself of their truth in the course of the investigation which follows. In the first place certain simple constructions, the drawing and producing of a straight line, and the drawing of a circle, must be assumed to be possible, and with the constructions the existence of such things as straight lines and circles ; and besides this we must lay down some postulate to form the basis of the theory of parallels." It is true that the admission of the 4th postulate that all right angles are equal still presents a difficulty to which we shall have to recur. There is of course no foundation for the idea, which has found its way into many text-books, that " the object of the postulates is to declare that the only instruments the use of which is permitted in geometry are the rule and compass'." § 4. THEOREMS AND PROBLEMS. " Again the deductions from the first principles," says Proclus, "are divided into problems and theorems, the former embracing the 1 Proclus, pp. i8j, 11—183, '3- * Anal. past. 1. aj, 86* 33—35. 8 Cf. Lardner's Euclid : aha Todhunter. ch. ix. 5 4 ] THEOREMS AND PROBLEMS i*5 generation, division, subtraction or addition of figures, and generally the changes which are brought about in them, the latter exhibiting the essential attributes of each 1 ," " Now, of the ancients, some, like Speusippus and Amphinomus, thought proper to call them all theorems, regarding the name of theorems as more appropriate than that of problems to theoretic sciences, especially as these deal with eternal objects. For there is no becoming in things eternal, so that neither could the problem have any place with them, since it promises the generation and making of what has not before existed, e.g. the construction of an equilateral triangle, or the describing of a square on a given straight line, or the placing of a straight line at a given point. Hence they say it is better to assert that all (propositions) are of the same kind, and that we regard the generation that takes place in them as referring not to actual making but to knowledge, when we treat things existing eternally as if they were subject to becoming: in other words, we may say that everything is treated by way^ of theorem and not by way of problem* (irdvra 6eapr)fjiaTiKw oX\* ov TrpQJ3\iifiaTtic<io<i \etfi0 ui' hit (tat). " Others on the contrary, like the mathematicians of the school of Menaechmus, thought it right to call them all problems, describing their purpose as twofold, namely in some cases to furnish (yropi- ffatr&at) the thing sought, in others to take a determinate object and see either what it is, or of what nature, or what is its property, or in what relations it stands to something else. "In reality both assertions are correct. Speusippus is right because the problems of geometry are not like those of mechanics, the latter being matters of sense and exhibiting becoming and change of every sort. The school of Menaechmus are right also because the discoveries even of theorems do not arise without an issuing- forth into matter, by which I mean intelligible matter. Thus forms going out into matter and giving it shape may fairly be said to be like processes of becoming. For we say that the motion of our thought and the thro wing-out of the forms in it is what produces the figures in the imagination and the conditions subsisting in them. It is in the imagination that constructions, divisions, placings, applications, additions and subtractions (take place), but everything in the mind is fixed and immune from becoming and from every sort of change 1 ." " Now those who distinguish the theorem from the problem say that every problem implies the possibility, not only of that which is predicated of its subject-matter, but also of its opposite, whereas every theorem implies the possibility of the thing predicated but not of its opposite as well. By the subject-matter I mean the genus which is the subject of inquiry, for example, a triangle or a square or a circle, and by the property predicated the essential attribute, as equality, section, position, and the like. When then any or-e ' Proclus, p, 77, 7— n. * Hid. pp. 77, 15—78, 8. * iHd, pp. j8, S — 79, 1. i26 INTRODUCTION [ch. ix, $4 enunciates thus, To inscribe an equilateral triangle in a circle, he states a problem ; for it is also possible to Inscribe in it a triangle which is not equilateral. Again, if we take the enunciation On a given limited straight line to construct an equilateral triangle, this is a problem ; for it is possible also to construct one which is not equi- lateral. But, when any one enunciates that In isosceles triangles tlu angles at the base are equal, we must say that he enunciates a theorem ; for it is not also possible that the angles at the base of isosceles triangles should be unequal. It follows that, if any one were to use the form of a problem and say In a semicircle to describe a right angle, he would be set down as no geometer. For every angle in a semi- circle is right 1 .'' '' Zenodotus, who belonged to the succession of Oenoptdes, but was a disciple of Andron, distinguished the theorem from the problem by the fact that the theorem inquires what is the property predicated of the subject-matter in it, but the problem what is the cause of what effect (rivoi Sirros ri icrriv). Hence too Posidonius denned the one (the problem) as a proposition in which it is inquired whether a thing exists or not (et ffo-rte t) /*■$), the other (the theorem*) as a proposition in which it is inquired what (a thing) is or of what nature (ri itrrtv fj •irolov Tt) ; and he said that the theoretic proposition must be put in a declaratory form, e.g., Any triangle has two sides (together) greater than the remaining side and In any isosceles triangle the angles at the base are equal, but that we should state the problematic proposition as If inquiring whether it is possible to construct an equilateral triangle upon such and such a straight line. For there is a difference between inquiring absolutely and indeterminately (a-n-Xaq re KaX dopl<rrto<i) whether there exists a straight line from such and such a point at right angles to such and such a straight line and investigating which is the straight line at right angles*." "That there is a certain difference between the problem and the theorem is clear from what has been said ; and that the Elements of Euclid contain partly problems and partly theorems will be made manifest by the individual propositions, where Euclid himself adds at the end of what is proved in them, in some cases, 'that which it was required to do,' and in others, ' that which it was required to prove, 1 the latter expression being regarded as characteristic of theorems, in spite of the fact that, as we have said, demonstration is found in problems also. In problems, however, even the demonstration is for the purpose of (confirming) the construction : for we bring in the demonstration in order to show that what was enjoined has been done ; whereas in theorems the demonstration is worthy of study for its own sake as being capable of putting before us the nature of the thing sought. And you will find that Euclid sometimes interweaves theorems with problems and employs them in turn, as in the first * Pioclus, pp. 79, n^So, 5. * In the text we have ri &t rpiflX^fia answering to ri pin without substantive : Trp&p\i)jin was obvious]; inserted in error, * Prochis, pp. 80, is — 8i, 4. ch. ix. 54] THEOREMS AND PROBLEMS ia? book, while at other times he makes one or other preponderate. For the fourth book consists wholly of problems, and the fifth of theorems 1 ." Again, in his note on Eucl. I. 4, Proclus says that Carpus, the writer on mechanics, raised the question of theorems and problems in his treatise on astronomy. Carpus, we are told, " says that the class of problems is in order prior to theorems. For the subjects, the properties of which are sought, are discovered by means of problems. Moreover in a problem the enunciation is simple and requires no skilled intelligence; it orders you plainly to do such and such a thing, to construct an equilateral triangle, or, given two straight lines, to cut off from the greater (a straight line) equal to the lesser, and what is there obscure or elaborate in these things ? But the enunciation of a theorem is a matter of labour and requires much exactness and scientific judgment in order that it may not turn out to exceed or fail short of the truth ; an example is found even in this proposition (I. 4), the first of the theorems. Again, in the case of problems, one general way has been discovered, that of analysis, by following which we can always hope to succeed ; it is this method by which the more obscure problems are investigated. But, in the case of theorems, the method of setting about them is hard to get hold of since ' up to our time,' says Carpus, ' no one has been able to hand down a general method for their discovery. Hence, by reason of their easiness, the class of problems would naturally be more simple/ After these distinctions, he proceeds : ' Hence it is that in the Elements too problems precede theorems, and the Elements begin from them ; the first theorem is fourth in order, not because the fifth 11 is proved from the problems, but because, even if it needs tor its demonstration none of the propositions which precede it, it was necessary that they should be first because they are problems, while it is a theorem. In fact, in this theorem he uses the common notions exclusively, and in some sort takes the same triangle placed in different positions ; the coincidence and the equality proved thereby depend entirely upon sensible and distinct apprehension. Nevertheless, though the demon- stration of the first theorem is of this character, the problems properly preceded it, because in general problems are allotted the order of precedence*.'" Proclus himself explains the position of Prop. 4 after Props, i- — 3 as due to the fact that a theorem about the essential properties of triangles ought not to be introduced before we know that such a thing as a triangle can be constructed, nor a theorem about the equality of sides or straight lines until we have shown, by constructing them, that there can be two straight lines which are equal to one another 4 . It is plausible enough to argue in this way that Props. 2 and 3 at all events should precede Prop, 4. And Prop, 1 is used in 1 Proclus, p. Si, 5 — is. 9 ri> ri^Trzav. This should apparently be the fourth because in the next words it is implied that none of the first three propositions ate required in proving it. * Proclus, pp. 141, it) — 543, 11. * ibid. pp. 133, 11— 1134, 6. 1*8 INTRODUCTION [ch. ix. §4 Prop. 2, and must therefore precede it But Prop. 1 showing how to construct an equilateral triangle on a given base is not important, in relation to Prop. 4, as dealing with the " production of triangles " in general : for it is of no use to say, as Proclus does, that the construc- tion of the equilateral triangle is " common to the three species (of triangles) 1 ," as we are not in a position to know this at such an early stage. The existence of triangles in general was doubtless assumed as following from the existence of straight lines and points in one plane and from the possibility of drawing a straight line from one point to another. Proclus does not however seem to reject definitely the view of Carpus, for he goes on* : "And perhaps problems are in order before theorems, and especially for those who need to ascend from the arts which are concerned with things of sense to theoretical investigation. But in dignity theorems are prior to problems.... It is then foolish to blame Geminus for saying that the theorem is more perfect than the problem. For Carpus himself gave the priority to problems in respect of order, and Geminus to theorems in point of more perfect dignity," so that there was no real inconsistency between the two. Problems were classified according to the number of their possible solutions. Amphinomus said that those which had a unique solution (itowa^tSt) were called " ordered " (the word has dropped out in Proclus, but it must be rerayftiva, in contrast to the third kind, aTattra) ; those which had a definite number of solutions " inter- mediate" (fiea-a); and those with an infinite variety of solutions "un- ordered " (ajaMTa)'. Proclus gives as an example of the last the problem To divide a given straight line into three parts in continued proportion*. This is the same thing as solving the equations %+y+s=a, xs =_y i . Proclus' remarks upon the problem show that it was solved, like all quadratic equations, by the method of " application of areas." The straight line a was first divided into any two parts, (jr-M)and_y, subject to the sole limitation that (x + s) must not be less than 2y, which limitation is the Stopto-fws, or condition of possibility. Then an area was applied to (x + z), or («— y), "falling short by a square figure" {eWetirov tXhei rerpaywy) and equal to the square on y. This determines x and z separately in terms of a and y. For, if b be the side of the square by which the area (i.e. rectangle) "falls short," we have {{a —y) — z\z ™j**, whence 2z «= (a —y) ± n/[(a —yf — 4y 3 }. And y may be chosen arbitrarily, provided that it is not greater than 0/3. Hence there are an infinite number of solutions. If y = a(i, then, as Proclus remarks, the three parts are equal. Other distinctions between different kinds of problems are added by Proclus. The word " problem," he says, is used in several senses. In its widest sense it may mean anything " propounded " (irpojewo- pi-i>ov), whether for the purpose of instruction {paBrio-em) or construc- tion (■n-oojo'fci)?). (In this sense, therefore, it would include a theorem.) 1 Proclus, p. 13^, Ji. ■ ibid, p. 543, 11—15. ■ ibid, p. »jo> 7 — 13. * ibid. pp. no, 16 — 111, 6. ch. ix. §4] THEOREMS AND PROBLEMS 139 But its special sense in mathematics is that of something "propounded with a view to a theoretic construction 1 ." Again you may apply the term (in this restricted sense) even to something which is impossible, although it is more appropriately used of what is possible and neither asks too much nor contains too little in the shape of data. According as a problem has one or other of these defects respectively, it is called (1) a problem in excess (irXeovdgov) or (2) a deficient problem (iK\nre<; irp60\T}p.a), The problem in excess (1) is of two kinds, (a) a problem in which the properties of the figure to be found are either inconsistent (atrii p/Sara) or non-existent {avvirapicTa), in which case the problem is called impossible, or (b) a problem in which the enunciation is merely redundant : an example of this would be a problem requiring us to construct an equilateral triangle with its vertical angle equal to two-thirds of a right angle ; such a problem is possible and is called "more than a problem '' (p.el&v 7j Trp6ft\T)fta). The deficient problem (2) is similarly called " less than a problem " (tKaatrov rj Trpdft\i)f*a), its characteristic being that something has to be added to the enunciation in order to convert it from indeterminateness (aopttr-ria) to order (raf «) and scientific deter- minateness (Spas evta-TtjtioviKos) : such would be a problem bidding you " to construct an isosceles triangle," for the varieties of isosceles triangles are unlimited. Such "problems" are not problems in the proper sense {/cupim<! \cy6ftei/a vpofiXjjpaTa), but only equivocally". § 5. THE FORMAL DIVISIONS OF A PROPOSITION. "Every problem," says Proclus', "and every theorem which is complete with all its parts perfect purports to contain in itself all of the following elements: enunciation (TrpoVao-is), setting-out (e*0Wt<:), definition or specification (Stop tvr fiat), construction or machinery (xaratrKt utj), proof (airoSeifiv), conclusion (uvp/!repai7jt,a). Now of these the enunciation states what is given and what is that which is sought, the perfect enunciation consisting of both these parts. The setting-out marks off what is given, by itself, and adapts it before- hand for use in the investigation. The definition or specification states separately and makes clear what the particular thing is which is sought. The construction or machinery adds what is wanting to the datum for the purpose of finding what is sought .The proof draws the required inference by reasoning scientifically from acknowledged facts. The conclusion reverts again to the enunciation, confirming what has been demonstrated. These are all the parts of problems and theorems, but the most essential and those which are found in all are enunciation, proof, conclusion. For it is equally necessary to know beforehand what is sought, to prove this by means of the intermediate steps, and to state the proved fact as a conclusion ; it is impossible to dispense with any of these three things. The remaining parts are often brought in, but are often left out as serving no purpose. 1 proclus, p. aii, J— 1 r. a ibid. pp. Mr, 13 — 111, i+. * ibid. pp. 503, i — ]o+, 13 ; 1O4, 13 — 105, 8. 130 INTRODUCTION [ch. rx. js Thus there is neither setting-out nor definition in the problem of constructing an isosceles triangle having each of the angles at the base double of the remaining angle, and in most theorems there is no construction because the setting'Oiit suffices without any addition for proving the required property from the data. When then do we say that the setting-out is wanting ? The answer is, when there is nothing given in the enunciation ; for, though the enunciation is in general divided into what is given and what is sought, this is not always the case, but sometimes it states only what is sought, i.e. what must be known or found, as in the case of the problem just mentioned. That problem does not, in fact, state beforehand with what datum we are to construct the isosceles triangle having each of the equal angles double of the remaining angle, but (simply) that we are to find such a triangle When, then, the enuncia- tion contains both (what is given and what is sought), in that case we find both definition and setting-out, but, whenever the datum is wanting, they too are wanting. For not only is the setting-out concerned with the datum, but so is the definition also, as, in the absence of the datum, the definition will be identical with the enunciation. In fact, what could you say in defining the object of the aforesaid problem except that it is required to find an isosceles triangle of the kind referred to ? But that is what the enunciation stated. If then the enunciation does not include, on the one hand, what is given and, on the other, what is sought, there is no setting-out in virtue of there being no datum, and the definition is left out in order to avoid a mere repetition of the enunciation." The constituent parts of an Euclidean proposition will be readily identified by means of the above description. As regards the defi- nition or specification (S*opt<r/ios) it is to be observed that we have here only one of its uses. Here it means a closer definition or descrip- tion of the object aimed at, by means of the concrete lines or figures set out in the eK&eai<; instead of the general terms used in the enun- ciation ; and its purpose is to rivet the attention better, as Proclus indicates in a later passage (rpoirov rtvk irpotrextia? itrrlv aiVioe o The other technical use of the word to signify the limitations to which the possible solutions of a problem are subject is also described by Proclus, who speaks of hiopio-fioi determining " whether what is sought is impossible or possible, and how far it is practicable and in how many ways 1 " ; and the Biopur/ioit in this sense appears in Euclid as well as in Archimedes and Apollonius. Thus we have in Eucl. I, 22 the enunciation " From three straight lines which are equal to three given straight lines to construct a triangle," followed imme- diately by the limiting condition ($tap«rfi6<;). " Thus two of the straight lines taken together in any manner must be greater than the remaining one." Similarly in VI. 28 the enunciation "To a given straight line to apply a parallelogram equal to a given rectilineal 1 Proclus, p. ?o8, ?i, * ibid. \i. loa, 3. ch. ix. f 5] FORMAL DIVISIONS OF A PROPOSITION 131 figure and falling short by a parallelogrammic figure similar to a given one " is at once followed by the necessary condition of possi- bility: "Thus the given rectilineal figure must not be greater than that described on half the line and similar to the defect." Tannery supposed that, in giving the other description of the Biopta/tos as quoted above, Proclus, or rather his guide, was using the term incorrectly. The Btopta-fios in the better known sense of the determination of limits or conditions of possibility was, we are told, invented by Leon. Pappus uses the word in this sense only. The other use of the term might, Tannery thought, be due to a confusion occasioned by the use of the same words (Bu &}) in introducing the parts of a proposition corresponding to the two meanings of the word Btopur/io^ 1 . On the other hand it is to be observed that Eutocius distinguishes clearly between the two uses and implies that the differ- ence was well known 1 . The SiopuTft.6^ in the sense of condition of possibility follows immediately on the enunciation, is even part of it ; the &iopitry.6s in the other sense of course comes immediately after the setting-out. Proclus has a useful observation respecting the conclusion of a proposition 3 . " The conclusion they are accustomed to make double in a certain way : 1 mean, by proving it in the given case and then drawing a general inference, passing, that is, from the partial con- clusion to the general. For, inasmuch as they do not make use of the individuality of the subjects taken, but only draw an angle or a straight line with a view to placing the datum before our eyes, they consider that this same fact which is established in the case of the particular figure constitutes a conclusion true of every other figure of the same kind. They pass accordingly to the general in order that we may not conceive the conclusion to be partial. And they are justified in so passing, since they use for the demonstration the par- ticular things set out, not qud particulars, but qua typical of the rest For it is not in virtue of such and such a size attaching to the angle which is set out that I effect the bisection of it, but in virtue of its being rectilineal and nothing more. Such and such size is peculiar to the angle set out, but its quality of being rectilineal is common to all rectilineal angles. Suppose, for example, that the given angle is a right angle. If then I had employed in the proof the fact of its being right, I should not have been able to pass to every species of recti- lineal angle ; but, if I make no use of its being right, and only consider it as rectilineal, the argument will equally apply to rectilineal angles in general." L La Giotnltru graque, p. 149 note. Where 5ti 5% introduces the closer description of the pioblem we may translate, "it is then requited'' or "thus it is required" (to construct etc); when it introduces the condition of possibility we may translate "thus it is necessary etc." Heiberg originally wrote Sti 9i in the latter sense in I. 21 on the authority of Proclus and Eutocius, and against that of the mss. Later, on the occasion of XI. 13, he observed that he should have followed the MSS. and written Sti i5?j which he found to be, after all, the right reading in Eutocius (Apollonius, ed. Heiberg, 11. p. 178). Je i Hi is also the expression used by Diophantus for introducing conditions of possibility. a See the passage of Eutocius referred to in last note. 3 Proclus, p. 307, 4 — 1$. i 3 a INTRODUCTION [ch. ix. §6 I 6. OTHER TECHNICAL TERMS. I. Things said to be given. Proclus attaches to his description of the formal divisions of a proposition an explanation of the different senses in which the word give ft or datum (SeSo^evoi') is used in geometry. " Everything that is given is given in one or other of the following ways, in position, in ratio, in tnagnitude, or in species. The point is given in position only, but a line and the rest may be given in all the senses 1 ." The illustrations which Proclus gives of the four senses in which a thing may be given are not altogether happy, and, as regards things which are given in position, in magnitude, and in species, it is best, I think, to follow the definitions given by Euclid himself in his book of Data. Euclid does not mention the fourth class, things given in ratio, nor apparently do any of the great geometers. (i) Given in position really needs no definition ; and, when Euclid says {Data, Def 4) that " Points, lines and angles are said to be given in position which always occupy the same place," we are not really the wiser. (3) Given in tnagnitude is defined thus {Data, Def. 1): "Areas, lines and angles are called given in magnitude to which we can find equals." Proclus' illustration is in this case the following : when, he says, two unequal straight lines are given from the greater of which we have to cut off a straight line equal to the lesser, the straight lines are obviously given in magnitude, " for greater and less, and finite and infinite are predications peculiar to magnitude." But he does not explain that part of the implication of the term is that a thing is given in magnitude only, and that, for example, its position is not given and is a matter of indifference (3) Given in species. Euclid's definition {Data, Def. 3) is: " Rectilineal figures are said to be given in species in which the angles are severally given and the ratios of the sides to one another are given." And this is the recognised use of the term (cf. Pappus, passim) Proclus uses the term in a much wider sense for which I am not aware of any authority. Thus, he says, when we speak of (bisect- ing) a given rectilineal angle, the angle is given in species by the word rectilineal, which prevents our attempting, by the same method, to bisect a curvilineal angle I On Eucl. 1. 9, to which he here refers, he says that an angle is given in species when e.g. we say that it is right or acute or obtuse or rectilineal or " mixed," but that the actual angle in the proposition is given in species only. As a matter of fact, we should say that the actual angle in the figure of the proposition is given in magnitude and not in species, part of the implication of given in species being that the actual magnitude of the thing given in species is indifferent ; an angle cannot be given in species in this sense at all. The confusion in Proclus' mind is shown when, after saying that a right angle is given in species, he describes a third of a right angle as given in magnitude. 1 Proclus, p, Wj, 13—15. ch. ix. §6] OTHER TECHNICAL TERMS 133 No better example of what is meant by given in species, in its proper sense, as limited to rectilineal figures, can be quoted than the given parallelogram in Eucl. VI. 28, to which the required parallelo- gram has to be made similar ; the former parallelogram is in fact given in species, though its actual size, or scale, is indifferent (4) Given in ratio presumably means something which is given by means of its ratio to some other given thing. This we gather from Proclus' remark (in his note on 1. 9) that an angle may be given in ratio " as when we say that it is double and treble of such and such an angle or, generally, greater and less." The term, however, appears to have no authority and to serve no purpose. Proclus may have derived it from such expressions as "in a given ratio" which are common enough. 2. Lemma. "The term lemma" says Proclus 1 , "is often used of any proposition which is assumed for the construction of something else : thus it is a common remark that a proof has been made out of such and such lemmas. But the special meaning of lemma in geometry is a proposition requiring confirmation. For when, in either construction or demonstration, we assume anything which has not been proved but requires argument, then, because we regard what has been assumed as doubtful in itself and therefore worthy of investigation, we call it a lemma*, differing as it does from the postulate and the axiom in being matter of demonstration, whereas they are immediately taken for granted, without demonstration, for the purpose of confirming other things. Now in the discovery of lemmas the best aid is a mental aptitude for it. For we may see many who are quick at solutions and yet do not work by method ; thus Cratistus in our time was able to obtain the required result from first principles, and those the fewest possible, but it was his natural gift which helped him to the discovery. 1 Proclus, pp. 111, 1 — m, 4. * It would appear, says Tannery (p. 1 ji «.), that Gem in us understood a lemma as being simply \atxflwt6f££vov t something assumed (cf. Lhe passage of Proclus, p. 73, 4, relating to Mcnaechmus* vie* of eUmertis) : hence we cannot consider ourselves authorised in attributing to Geminus the more technical definition of the term here given by Proclus, according to which it is only used of propositions not proved beforehand. This view of a lemma must be considered as relatively modern. It seems to have had its origin in an imperfection of method. In the course of a demonstration it was necessary to assume a proposition which required proof, but the proof of which would, if inserted in lhe particular place, break the thread of the demonstration : hence it was necessary either to prove it beforehand as a preliminary proposition or to postpone it to be proved afterwards {<Sit i& inx^irtrai). when, after the time of Ge minus, the progress of original discovery in geometry was arrested, geometers occupied themselves with the study and elucidation of the works of the great mathematicians who had preceded them. This involved the investigation of propositions explicitly quoted or tacitly assumed in the great classical treatises; and naturally it was found that several such remained to be demonstrated, either because Lhe authors had omitted them as being easy enough to be left to the reader himself to prove, or because books in which they were proved had been lost in the meantime. Hence arose a class of complementary or auxiliary propositions which were called lemmas. Thus Pappus gives in his Book VII a collection of lemmas in elucidation of the treatises of Euclid and Apollonius included in the so-called "Treasury of Analysis " (r&rot AvtOwdtxevat), When Procltis goes on to distinguish three methods of discovering lemmas, analysis, division, and redurtio ad absurdum, he seems to imply that the principal business of contemporary geometers was the investigation of these auxiliary propositions. i 3 4 INTRODUCTION [ch. ix. §6 Nevertheless certain methods have been handed down. The finest is the method which by means of analysis carries the thing sought up to an acknowledged principle, a method which Plato, as they say, com- municated to Leodamas 1 , and by which the latter, too, is said to have discovered many things in geometry. The second is the method of division*, which divides into its parts the genus proposed for con- sideration and gives a starting-point for the demonstration by means of the elimination of the other elements in the construction of what is proposed, which method also Plato extolled as being of assistance to all sciences. The third is that by means of the reductio ad absurdntn, which does not show what is sought di recti yi but refutes its opposite and discovers the truth incidentally.'' 3. Case. "The case* (irTmtTK)," Proclus proceeds*, "announces different ways of construction and alteration of positions due to the transposition of points or lines or planes or solids. And, in general, all its varieties are seen in the figure, and this is why it is called case, being a trans- position in the construction." 4. Porism. " The term porism is used also of certain problems such as the Porisms written by Euclid. But it is specially used when from what has been demonstrated some other theorem is revealed at the same time without our propounding it, which theorem has on this very account been called a porism (corollary) as being a sort of incidental gain arising from the scientific demonstration ." Cf. the note on 1. 15, 1 This passage and another from Diogenes Laertiiis (ur. 54, p. 74 ed. Coliet) to the effect that "He [Plato] explained [tttnj-rfifaTO} to Leodamas of Thasos the method of inquiry by analysis " have been commonly understood as ascribing to Plato the invention of the method of analysis ; but Tannery points out forcibly (pp. 1 13, 113) bow difficult it is to explain in what Plato's discovery could have consisted if analysis be taken in the sense attributed to it in Pappus, where we can see no more than a series of successive, reductions of a problem until it is finally reduced to a known problem. On the other hand, Proclus' words about carrying up the thing sought to " an acknowledged principle " suggest that what he had in mind was the process described at the end of Book vr of the Republic by which the dialec- tician (unlike the mathematician) uses hypotheses as stepping-stones up to a principle which is not hypothetical, and then is able to descend step by step verifying every one of the hypotheses by which he ascended. This description does not of course refer to mathematical analysis, but it may have given rise to the idea that analysis was Plato's discovery, since analysis and synthesis following each other are related in the same way as the upward and the downward progression in the dialectician's intellectual method- And it may be that Plato's achievement was to observe the importance, from the point of view of logical rigour, of the confirmatory synthesis following analysis, and to regularise in this way and elevate into a completely irrefragable method the partial and uncertain analysis upon which the works of his predecessors depended. * Here again the successive biparlilions of genera into species such as we find in the Sephist and Republic have very little to say to geometry, and the very fact that they are here mentioned side by side with analysis suggests that Proclus confused the latter with the philosophical method of Hep. vi. 3 Tannery rightly remarks (p. 151) that the subdivision of a theorem or problem into several cases is foreign to the really classic form ; the ancients preferred, where necessary, to multiply enunciations. As, however, some omissions necessarily occurred, the writers of lemmas naturally added separate cases, which in some instances found their way into the text. A good example is Euclid I. 7, the second case of which, as it appears in our text- books, was interpolated. On the commentary of Proclus on this proposition Th, Taylor rightly remarks that " Euclid everywhere avoids a multitude of cases." * Proclus, p. 11a, 5 — 11. a Tannery notes however that, so far from distinguishing his corollaries from the con- ch. ix. §6] OTHER TECHNICAL TERMS 135 5. Objection. " The objection (eWrao-i?) obstructs the whole course of the argu- ment by appearing as an obstacle (or crying ' halt,' d-n-avrao-a) either to the construction or to the demonstration. There is this difference between the objection and the case, that, whereas he who propounds the case has to prove the proposition to be true of it, he who makes the objection does not need to prove anything : on the contrary it is necessary to destroy the objection and to show that its author is saying what is false'." That is, in general the objection endeavours to make it appear that the demonstration is not true in every case ; and it is then necessary to prove, in refutation of the objection, either that the supposed case is impossible, or that the demonstration is true even for that case. A good instance is afforded by EucL I. 7, The text- books give a second case which is not in the original text of Euclid. Proclus remarks on the proposition as given by Euclid that the objection may conceivably be raised that what Euclid declares to be impossible may after all be possible in the event of one pair of stiaight lines falling completely within the other pair. Proclus then refutes the objection by proving the impossibility in that case also. His proof then came to be given in the text-books as part of Euclid's proposition. The objection is one of the technical terms in Aristotle's logic and its nature is explained in the Prior Analytics''. "An objection is a proposition contrary to a proposition.,.. Objections are of two sorts, general or partial.... For when it is maintained that an attribute belongs to every (member of a class), we object either that it belongs to none (of the class) or that there is some one (member of the class) to which it does not belong." 6. Reduction. This is again an Aristotelian term, explained in the Prior Analytics*. It is well described by Proclus in the following passage : " Reduction (diraywyq) is a transition from one problem or theorem to another, the solution or proof of which makes that which is pro- pounded manifest also. For example, after the doubling of the cube had been investigated, they transformed the investigation into another upon which it follows, namely the finding of the two means ; and from that time forward they inquired how between two given straight lines two mean proportionals could be discovered. And Tney say that the first to effect the reduction of difficult constructions was Hippocrates of Chios, who also squared a lune and discovered many other things in geometry, being second to none in ingenuity as regards constructions'." elusions of his propositions, Euclid inserts them before the closing words " (being) what it was required to do'' or M to prove." In fact the porism- corollary is with Euclid rather a modified form of the regular conclusion than a separate proposition. 1 Proclus, p. an, 18 — 33. * Anal, prior. 11. 16, 69 a 37.- * ibid. 11. 15, 69 a 30. * Proclus, pp. an, 34— 113, 11. This passage has frequently been taken as crediting Hippocrates with the discovery of the method of geometrical reduction: ef. Taylor (Transla- tion of Proclus, 11. p. 16), Allman (p. 41 4., 50), Gow (pp. 169, 170), As Tannery remarks (p. no), if the particular reduction of the duplication problem to that of the two means is 136 INTRODUCTION [ch. IX. {6 7. Re duct io ad absurdum. This is variously called by Aristotle " reductio ad absurdum" ft) «*s rr) aZvvarav airaycoyi})', " proof per impossible " (17 Bia toS dZvvarov Setfw or a-n-o&etfcy, " proof leading to the impossible " (r) Wc to rtSufaroy ayova-a airoBeti is) 1 . It is part of " proof (starting) from a hypothesis' " («'f tiTrofeo-ew?). " All (syllogisms) which reach the conclusion per impossibile reason out a conclusion which is false, and they prove the original contention (by the method starting) from a hypothesis, when something impossible results from assuming the contradictory of the original contention, as, for example, when it is proved that the diagonal (of a square) is incommensurable because, if it be assumed commensurable, it will follow that odd (numbers) are equal to even (numbers) 1 ." Or again, "proof (leading) to the impossible differs from the direct (£ec*r(«ri}t) in that it assumes what it desires to destroy [namely the hypothesis of the falsity of the conclusion] and then reduces it to something admittedly false, whereas the direct proof starts from premisses admittedly true'." Proclus has the following description of the reductio ad absurdum. " Proofs by reductio ad absurdum in every case reach a conclusion manifestly impossible, a conclusion the contradictory of which is admitted. In some cases the conclusions are found to conflict with the common notions, or the postulates, or the hypotheses (from which we started); in others they contradict propositions previously estab- lished 7 ". .."Every reductio ad absurdum assumes what conflicts with the desired result, then, using that as a basis, proceeds until it arrives at an admitted absurdity, and, by thus destroying the hypothesis, establishes the result originally desired. For it is necessary to under- stand generally that all mathematical arguments either proceed from the first principles or lead back to them, as Porphyry somewhere says. And those which proceed from the first principles are again of two kinds, for they start either from common notions and the clearness of the self-evident alone, or from results previously proved ; while those which lead back to the principles are either by way of assuming the principles or by way of destroying them. Those which assume the principles are called analyses, and the opposite of these are syntheses — for it is possible to start from the said principles and to proceed in the regular order to the desired conclusion, and this process is syn- thesis — while the arguments which would destroy the principles are the first noted in history, it is difficult to suppose that it was really the first ; for Hippocrates must have found instances of it in the Pythagorean geometry. Bret Schneider, 1 think, comes nearer the truth when he boldly (p. 99) translates: "This reduction <tf tht aforesaid con- struct 'iott is said to have been first given by Hippocrates." The words are vpQrroy Si $a<n ntr &-ropovtt4vup StaypaftfiAruv r^v iira-ytoy^v ro^iratrffai, which must, literally, be translated as in the text above ; but, when Proclus speaks vaguely of " di ffi cult constructions," he probably means to say simply that " this first recorded instance of a reduction of a difficult construction is attributed to Hippocrates." 1 Aristotle, Anal, prior. I. 7, 19 b 5 ; 1. 44, 50 a 30. " ibid. 1. it, 39 b 35 ; 1. 19, +5 & 35. 3 Anal. post. I, 14, 85 a 16 etc. * Anal, prior, \. 13, 40 b tj. * Anal, prior. I. 53, 41 a 14, • ibid. n. 14, 61 b 19. ' Proclus, p. 1J4, Ji — 17. ch. ix. $6] OTHER TECHNICAL TERMS 137 called reductiones ad absurdum. For it is the function of this method to upset something admitted as clear 1 ." 8. Analysis and Synthesis. It will be seen from the note on Eucl. xin. i that the mss. of the Elements contain definitions of Analysis and Synthesis followed by alternative proofs of xill. i — 5 aftet that method. The definitions and alternative proofs are interpolated, but they have great historical interest because of the possibility that they represent an ancient method of dealing with these propositions, anterior to Euclid. The propositions give properties of a line cut " in extreme and mean ratio," and they are preliminary to the construction and comparison of the five regular solids. Now Pappus, in the section of his Collection dealing with the latter subject", says that he will give the comparisons between the five figures, the pyramid, cube, octahedron, dodecahedron and icosahedron, which have equal surfaces, " not by means of the so-called analytical inquiry, by which some of the ancients worked out the proofs, but by the synthetical method 1 ...." The conjecture of Bretschneider that the matter interpolated in Eucl. XIII. is a survival of investiga- tions due to Eudoxus has at first sight much to commend it 4 . In the first place, we are told by Proclus that Eudoxus "greatly added to the number of the theorems which Plato originated regarding the sectim,&nd employed in them the method of analysis'." It is obvious that " the section " was some particular section which by the time of Plato had assumed great importance ; and the one section of which this can safely be said is that which was called the " golden section," namely, the division of a straight line in extreme and mean ratio which appears in Eucl. II. 1 1 and is therefore most probably Pytha- gorean. Secondly, as Cantor points out 8 , Eudoxus was the founder of the theory of proportions in the form in which we find it in Euclid v., VI., and it was no doubt through meeting, in the course of his investigations, with proportions not expressible by whole numbers that he came to realise the necessity for a new theory of proportions which should be applicable to incommensurable as well as commen- surable magnitudes. The "golden section" would furnish such a case. And it is even mentioned by Proclus in this connexion. He is explaining' that it is only in arithmetic that all quantities bear "rational" ratios (pyros Xoyos) to one another, while in geometry there are " irrational " ones (appqros) as well. " Theorems about sections like those in Euclid's second Book are common to both [arithmetic and geometry] except that in which the straight line is cut in extreme and mean ratio"." 1 Proclus, p, i5j, ' — '** s Pappus, v. p. 410 soq, * ibid. pp. +10, 17—411, i. * Bretschneider, p. ioB. See however Heiberg's recent suggestion {Paratipomcna zu Euklid in Hermts, , XXXVIII., 1W3) that the author was Heron. The suggestion is based on a comparison with the remarks on analysis and synthesis quoted from Heron by an-NairtzI (ed. Curtie, P- 89) at the beginning of his commentary on Eucl. Book II. On the whole, this suggestion commends itself to me more than that of Bretschneider. 5 Proclus, p. 67, 6. B Cantor, Gtsck. d. Math. I 3 , p. 34 j. 7 Proclus, p, 60, 7—9. B ibid. p. 60, [6 — 19. ij8 INTRODUCTION [ch. ix. }6 The definitions of Analysis and Synt Justs interpolated in Eucl. XIII. are as follows (I adopt the reading of B and V, the only in- telligible one, for the second). " Analysis is an assumption of that which is sought as if it were admitted < and the passage > through its consequences to something admitted (to be) true. " Synthesis is an assumption of that which is admitted < and the passage > through its consequences to the finishing or attainment of what is sought." The language is by no means clear and has, at the best, to be filled out. Pappus has a fuller account 1 : " The so-called ivakv&fMPW (' Treasury of Analysis ") is, to put it shortly, a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis. " Analysis then takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (yeyovos), and we inquire what it is from which this results, and again what is the ante- cedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (deaira\tv \vtrtr). " But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought ; and this we call synthesis. " Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical, (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted : then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (d), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted : if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in 1 Pappus, VII. pp. 63+ — 6. ch. ix. $6] OTHER TECHNICAL TERMS 139 reverse order to the analysis, but if {&) we come upon something admittedly impossible, the problem will also be impossible." The ancient Analysis has been made the subject of careful studies by several writers during the last half-century, the most complete being those of Hankel, Duhamel and Zeuthen.; others by Ofterdinger and Cantor should also be mentioned'. The method is as follows. It is required, let us say, to prove that a certain proposition A is true. We assume as a hypothesis that A is true and, starting from this we find that, if A is true, a certain other proposition B is true ; if B is true, then C ; and so on until we arrive at a proposition K which is admittedly true. The object of the method is to enable us to infer, in the reverse order, that, since K is true, the proposition A originally assumed is true. Now Aristotle had already made it clear that false hypotheses might lead to a conclusion which is true. There is therefore a possibility of error unless a certain precaution is taken. While, for example, B may be a necessary consequence of A, it may happen that A is not a necessary consequence of B, Thus, in order that the reverse inference from the truth of K that A is true may be logically justified, it is necessary that each step in the chain of inferences should be unconditionally convertible. As a matter of fact, a very large number of theorems in elementary geometry are unconditionally convertible, so that in practice the difficulty in securing that the successive steps shall be convertible is not so great as might be supposed. But care is always necessary. For example, as Hankel says 1 , a proposition may not be uncon- ditionally convertible in the form in which it is generally quoted. Thus the proposition "The vertices of all triangles having a common base and constant vertical angle lie on a circle " cannot be converted into the proposition that "All triangles with common base and vertices lying on a circle have a constant vertical angle"; for this is only true if the further conditions are satisfied (]) that the circle passes through the extremities of the common base and {2) that only that part of the circle is taken as the locus of the vertices which lies on one side of the base. If these conditions are added, the proposition is unconditionally convertible. Or again, as Zeuthen remarks 3 , K may be obtained by a series of inferences in which A or some other proposition in the series is only apparently used ; this would be the case e.g. when the method of modem algebra is being employed and the expressions on each side of the sign of equality have been inadvertently multiplied by some composite magnitude which is in reality equal to zero. Although the above extract from Pappus does not make it clear that each step in the chain of argument must be convertible in the case taken, he almost implies this in the second part of the definition jf Analysis where, instead of speaking of the consequences B, C... 1 lAx^.^ZurGtstkichlederMathetnatikinAlterihumMttdMUitiatter^ 1874, pp. 137 — 150; Duhamd, Damithodis dans la scititra de raittmnement. Part I., 3 ed., Paris, 1885, pp. 39 — 68 ; Zeutben, Gmkuktt der MtUhematik im Attcrtnm und Miltclalttr, 1896, pp. 92 — 104; Oflerdinger, Beitrdp zur Gisehkhte der gritchischm MathmnUii, Ulm, i860; Cantor, Geschkkte der Mathtmatiki l t , pp. 110 — i. 1 Hankel, p. 139. * Zeiuhen, p. 103. Mo INTRODUCTION [ch. ix. 5 6 successively following from A, he suddenly changes the expression and says that we inquire what it is (B)frem which A follows (A being thus the consequence of B, instead of the reverse), and then what (viz. C) is the antecedent cause of B; and in practice the Greeks secured what was wanted by always insisting on the analysis being confirmed by subsequent synthesis, that is, they laboriously worked backwards the whole way from K. to A, reversing the order of the analysis, which process would undoubtedly bring to light any flaw which had crept into the argument through the accidental neglect of the necessary precautions. Reductio ad absurdum a variety of analysis. In the process of analysis starting from the hypothesis that a proposition A is true and passing through B, C... as successive con- sequences we may arrive at a proposition K which, instead of being admittedly true, is either admittedly false or the contradictory of the original hypothesis A or of some one or more of the propositions B, C... intermediate between A and K. Now correct inference from a true proposition cannot lead to a false proposition ; and in this case there- fore we may at once conclude, without any inquiry whether the various steps in the argument are convertible or not, that the hypo- thesis A is false, for, if it were true, all the consequences correctly inferred from it would be true and no incompatibility could arise. This method of proving that a given hypothesis is false furnishes an indirect method of proving that a given hypothesis A is true, since we have only to take the contradictory of A and to prove that it is false. This is the method of reductio ad absurdum, which is therefore a variety of analysis. The contradictory of A, or not-A, will generally include more t han one case and, in order to prove its falsity, each of the cases must be separately disposed of: e.g., if it is desired to prove that a certain part of a figure is equal to some other part, we take separately the hypotheses (i) that it is greater, (2) that it is less, and prove that each of these hypotheses leads to a conclusion either admittedly false or contradictory to the hypothesis itself or to some one of its consequences. Analysis as applied to problems. It is in relation to problems that the ancient analysis has the greatest significance, because it was the one general method which the Greeks used for solving all "the more abstruse problems" (ra dera<f>G(TTepa twv TrpopKufiaTwvy '. We have, let us suppose, to construct a figure satisfying a certain set of conditions If we are to proceed at all methodically and not by mere guesswork, it is first necessary to "analyse" those conditions. To enable this to be done we must get them clearly in our minds, which is only possible by assuming all the conditions to be actually fulfilled, in other words, by supposing the problem solved. Then v;e have to transform those conditions, by all the means which practice in such cases has taught us to employ, into other conditions which are necessarily fulfilled if the original conditions are, and to continue this 1 Proclus, p. 141, 16, 17. CH. ix. $6] OTHER TECHNICAL TERMS 141 transformation until we at length arrive at conditions which we are in a position to satisfy 1 . In other words, we must arrive at some relation which enables us to construct a particular part of the figure which, it is true, has been hypothetically assumed and even drawn, but which nevertheless really requires to be found in order that the problem may be solved. From that moment the particular part of the figure becomes one of the data, and a fresh relation has to be found which enables a fresh part of the figure to be determined by means of the original data and the new one together. When this is done, the second new part of the figure also belongs to the data ; and we proceed in this way until all the parts of the required figure are found 1 . The first part of the analysis down to the point of discovery of a relation which enables us to say that a certain new part of the figure not belonging to the original data is given, Hankel calls the transformation ; the second part, in which it is proved that all the remaining parts of the figure are "given," he calls the resolution. Then follows the synthesis, which also consists of two parts, (1) the construction, in the order in which it has to be actually carried out, and in general following the course of the second part of the analysis, the resolution ; (2) the demonstration that the figure obtained does satisfy all the given conditions, which follows the steps of the first part of the analysis, the transformation, but in the reverse order. The second part of the analysis, the resolution, would be much facilitated and shortened by the existence of a systematic collection of Data such as Euclid's book bearing that title, consisting of propositions proving that, if in a figure certain parts or relations me. given, other parts or relations are also given. As regards the first part of the analysis, the trans- formation, the usual rule applies that every step in the chain must be unconditionally convertible ; and any failure to observe this condition will be brought to light by the subsequent synthesis. The second part, the resolution, can be directly turned into the construction since that only is given which can be constructed by the means provided in the Elements. It would be difficult to find a better illustration of the above than the example chosen by Hankel from Pappus*. Given a circle ABC and two points D, E external to it, to draw straight lines DB, KB from D, E to a point B on the circle such that, if DB, KB produced meet the circle again in C, A, AC shall be parallel A>DE. Analysis. Suppose the problem solved and the tangent at A drawn, meeting ED produced in F. (Part I. Transformation.) Then, since AC is parallel to DE, the angle at C is equal to the angle CDE. But, since FA is a tangent, the angle at C is equal to the angle FAE. Therefore the angle FAE is equal to the angle CDE, whence A, B, D, Fact concyclic. 1 Zcutben, p. 93. * Hankel, p. 141. ' Pappus, vn. pp. 630 — 1. 14* INTRODUCTION [ch. IX. 56 Therefore the rectangle AE, EB is equal to the rectangle FE, ED. (Part II, Resolution.) But the rectangle AE, EB is given, because it is equal to the square on the tangent from E. Therefore the rectangle FE, ED is given ; and, since ED is given, FE is given (in length). [Data, 57.J But FE is given in position also, so that F is also given. [Data, 27.] Now FA is the tangent from a given point F to a circle A BC given in position ; therefore FA is given in position and magnitude. [Data, 90.] And F is given ; therefore A is given. But £ is also given ; therefore the straight line AE is given in position, [Data, 26.] And the circle ABC is given in position ; therefore the point B is also given. [Data, 25.] But the points D, E are also given ; therefore the straight lines DB, BE are also given in position. Synthesis. (Part I. Construction) Suppose the circle ABC and the points D, E given. Take a rectangle contained by ED and by a certain straight line EF equal to the square on the tangent to the circle from E. From F draw FA touching the circle in A ; join ABE and then DB, producing DB to meet the circle at C. Join AC. 1 say then that A C is parallel to DE. (Part II. Demonstration.) Since, by hypothesis, the rectangle FE, ED is equal to the square on the tangent from E, which again is equal to the rectangle AE, EB, the rectangle AE, EB is equal to the rectangle FE, ED, Therefore A, B, D, F are concyclic, whence the angle FAE is equal to the angle BDE. But the angle FAE is equal to the angle ACB in the alternate segment ; therefore the angle A CB is equal to the angle BDE. Therefore AC Is parallel to DEk In cases where a &iopt<rft6<; is necessary, i.e. where a solution is only possible under certain conditions, the analysis will enable those conditions to be ascertained. Sometimes the Stopw/ioe is stated and proved at the end of the analysis, e.g. in Archimedes, On the Sphere and Cylinder, II. 7 ; sometimes it is stated in that place and the proof postponed till after the end of the synthesis, e.g. in the solution of the problem subsidiary to Oh the Sphere and Cylinder, II. 4, preserved in Eutocius' commentary on that proposition. The analysis should also enable us to determine the number of solutions of which the problem is susceptible. ch. ix. §7] THE DEFINITIONS 143 I 7. THE DEFINITIONS. General. "Real" and "Nominal" Definitions. It is necessary, says Aristotlej whenever any one treats of any whole subject, to divide the genus into its primary constituents, those which are indivisible in species respectively : e.g. number must be divided into triad and dyad ; then an attempt must be made in this way to obtain definitions, e.g. of a straight line, of a circle, and of a right angle 1 . The word for definition is 3po?. The original meaning of this word seems to have been " boundary," " landmark." Then we have it in Plato and Aristotle in the sense of standard or determining principle ("id quo alicuius rei natura constituitur vel definitur," Index Aristotelicus) 1 ; and closely connected with this is the sense of definition. Aristotle uses both JSpo? and 6pnrfia<t for definition, the former occurring more frequently in the Topics, the latter in the Metaphysics. Let us now first be clear as to what a definition does not do. There is nothing in connexion with definitions which Aristotle takes more pains to emphasise than that a definition asserts nothing as to the existence or non-existence of the thing defined. It is an answer to the question what a thing is (ri 4cm), and does not say that it is (5rt cart). The existence of the various things defined has to be proved, except in the case of a few primary things in each science, the existence of which is indemonstrable and must be assumed among the first principles of each science ; e.g. points and lines in geometry must be assumed to exist, but the existence of everything else must be proved This is stated clearly in the long passage quoted above under First Principles'. It is reasserted in such passages as the following. " The (answer to the question) what is a man and the fact that a man exists are different things'." " It is clear that, even according to the view of definitions now current, those who define things do not prove that they exist 1 ." " We say that it is by demonstration that we must show that everything exists, except essence (tl fii) ova-la eiij). But the existence of a thing is never essence; for the existent is not a genus. Therefore there must be demonstration that a thing exists. Thus, what is meant by triangle the geometer assumes, bat that it exists he has to prove 11 ." "Anterior knowledge of two sorts is necessary : for it is necessary to presuppose, with regard to some things, that they exist; in other cases it is necessary to understand what the thing described is, and in other cases it is necessary to do both. Thus, with the fact that one of two contradictories must be true, we must know that it exists (is true); 1 Anal, pest, tt. 13, 96 b ij. ' Of. Dt anitna, I. 1, 404 a g, where " breathing" is spoken of as the Spot of " life," and the many passages in the Politic! where the wurd is used to denote that which gives its special character to the several forms of government (virtue being the &poi of aristocracy, wealth of oligarchy, liberty of democracy, 130.4 a 10) ', Plato, fapuhtic, vju, 551 c * Anal. post. I. 10, 76 a 31 sqq. * ibid. IJ. 7, 91 b 10* * ibid. 91 b 19. ' ibid. 91 h 11 sqq. 144 INTRODUCTION [at. ix. § 7 of the triangle we must know that it means such and such a thing ; of the unit we must know both what it means and that it exists 1 ." What is here so much insisted on is the very fact which Mill pointed out in his discussion of earlier views of Definitions, where he says that the so-called real definitions or definitions of things do not constitute a different kind of definition from nominal definitions, or definitions of names ; the former is simply the latter plus something else, namely a covert assertion that the thing defined exists, "This covert assertion is not a definition but a postulate. The definition is a mere identical proposition which gives information only about the use of language, and from which no conclusion affecting matters of fact can possibly be drawn. The accompanying postulate, on the other hand, affirms a fact which may lead to consequences of every degree of importance. It affirms the actual or possible existence of Things possessing the combination of attributes set forth in the definition : and this, if true, may be foundation sufficient on which to build a whole fabric of scientific truth 1 ." This statement really adds nothing to Aristotle's doctrine' ; it has even the slight disadvantage, due to the use of the word "postulate" to describe "the covert assertion" in all cases, of not definitely pointing out that there are cases where existence has to be proved as distinct from those where it must be assumed. It is true that the existence of a definiend may have to be taken for granted provisionally until the time comes for proving it; but, so far as regards any case where existence must be proved sooner or later, the provisional assumption would be for Aristotle, not a postulate, but a hypothesis. In modern times, too, Mill's account of the true distinction between real and nominal definitions had been fully anticipated by Saccheri 1 , the editor of Euclides ab omni naevo vindicatas (1733), famous in the history of non-Euclidean geometry. In his Logica Demonstrativa (to which he also refers in his Euclid) Saccheri lays down the clear distinction between what he calls de- finitiones quid nominis or nominales, and definitiones quid rei or reales, namely that the former are only intended to explain the meaning 1 Anal. post. I. i, 71 a II sqq. s Mill's Syiltm of Logic, Bk. [. ch. viii. * It istrup ihat it was in opposition to *Mhe ideas of most of the Aristoteiian logicians" (rather than of Aristotle himself) that Mill laid such stress on his point of view. Cf. his observation : " We have already made, and shall often have to repeat, the remark, that the Ehilosophers who overthrew Realism by no means got rid of the consequences of Realism, ut retained long afterwards, in their own philosophy, numerous propositions which could only have a rational meaning as part of a Realistic system. It had been handed down from Anstotle, and probably from earlier times, as an obvious truth, that the science of geometry is deduced from definitions. This, so long as a definition was considered to be a proposition ' unfolding the nature of the thing,' did well enough. But Hobbes followed and rejected utterly the notion that a definition declares the nature of the thing, or does anything but state the meaning of a name ; yet he continued to affirm as broadly as any of his predecessors that the dpx&l, prituipia, or original premisses of mathematics, and even of all science, are definitions ; producing the singular paradox that systems of scientific truth, nay, all truths whatever at which we arrive by reasoning, are deduced from the arbitrary conventions of mankind concerning the signification of words." Aristotle was guilty of no such paradox ; on the contrary, he exposed it as plainly as did Mill. * This has been fully brought out in two papers by G. Vailati, La ttaria Ariitottiita del/a definiziow [Rivista di Filosofia s sciertse affini. 190;}, and Di un l opera dimtrUicata dtl P. Gsrolanto Satthtri (*' Logica Demonstrativa," 1097) (in Hivista Fifosefica, 1903). en. IX. 5 7] THE DEFINITIONS 145 that is to be attached to a given term, whereas the latter, besides declaring the meaning of a word, affirm at the same time the existence of the thing defined or, in geometry, the possibility of constructing it. The definitio quid nominis becomes a definitio quid rei " by means of a postulate, or when we come to the question whether the thing exists and it is answered affirmatively 1 ." Definitiones quid nominis are in them- selves quite arbitrary, and neither require nor are capable of proof; they are merely provisional and are only intended to be turned as quickly as possible into definitiones quid rei, either (1) by means of a postulate in which it is asserted or conceded that what is defined exists or can be constructed, e.g. in the case of straight lines and circles, to which Euclid's first three postulates refer, or (2) by means of a demonstration reducing the construction of the figure defined to the successive carrying-out of a certain number of those elementary constructions, the possibility of which is postulated. Thus definitiones quid rei are in general obtained as the result of a series of demonstrations. Saccheri gives as an instance the construction of a square in Euclid I. 46. Suppose that it is objected that Euclid had no right to define a square, as he does at the beginning of the Book, when it was not certain that such a figure exists in nature; the objection, he says, could only have force if, before proving and making the construction, Euclid had assumed the aforesaid figure as given. That Euclid is not guilty of this error is clear from the fact that he never presupposes the existence of the square as defined until after I. 46. Confusion between the nominal and the real definition as thus de- scribed, i.e. the use of the former in demonstration before it has been turned into the latter by the necessary proof that the thing defined exists, is according to Saccheri one of the most fruitful sources of illusory demonstration, and the danger is greater in proportion to the "complexity" of the definition, i.e. the number and variety of the attributes belonging to the thing defined. For the greater is the possibility that there may be among the attributes some that are incompatible, i.e. the simultaneous presence of which in a given figure can be proved, by means of other postulates etc. forming part of the basis of the science, to be impossible. The same thought is expressed by Leibniz also. " If," he says, " we give any definition, and it is not clear from it that the idea, which we ascribe to the thing, is possible, we cannot rely upon the demon- strations which we have derived from that definition, because, if that idea by chance involves a contradiction, it is possible that even con- tradictories may be true of it at one and the same time, and thus our demonstrations will be useless. Whence it is clear that definitions are not arbitrary. And this is a secret which is hardly sufficiently known'." Leibniz' favourite illustration was the " regular polyhedron with ten faces," the impossibility of which is not obvious at first sight. 1 "' Definitio quid neminit nata est evadere definitio quid rri per poiitttaium vel dum venitur ad quaestionem an at et respondetur affirmative. ' ' Opusottts ttfragmenUinidUtdiLtibmi, Paris, Mean, 1903, p. 4 ji. Quoted by Vailati. 146 INTRODUCTION [ch. ix. { 7 It need hardly be added that, speaking generally, Euclid's defini- tions, and his use of them, agree with the doctrine of Aristotle that the definitions themselves say nothing as to the existence of the things defined, but that the existence of each of them must be proved or (in the case of the " principles ") assumed. In geometry, says Aristotle, the existence of points and lines only must be as- sumed, the existence of the rest being proved. Accordingly Euclid's first three postulates declare the possibility of constructing straight lines and circles (the only " lines " except straight lines used in the Elements). Other things are defined and afterwards constructed and proved to exist : e.g. in Book I., Def. 20, it is explained what is meant by an equilateral triangle ; then (1. 1 ) it is proposed to construct it, and, when constructed, it is proved to agree with the definition. When a square is defined (1. Def. 22), the question whether such a thing really exists is left open until, in I. 46, it is proposed to construct it and, when constructed, it is proved to satisfy the definition 1 . Similarly with the right angle (I. Def. 10, and I. 11) and parallels (I. Def. 23, and I. 27—29). The greatest care is taken to exclude mere presumption and imagination. The transition from the sub- jective definition of names to the objective definition of things is made, in geometry, by means of constructions (the first principles of which are postulated), as in other sciences it is made by means of experience 1 . Aristotle's requirements in a definition. We now come to the positive characteristics by which, according to Aristotle, scientific definitions must be marked. First, the different attributes in a definition, when taken separately, cover more than the notion defined, but the combination of them does not Aristotle illustrates this by the " triad," into which enter the several notions of number, odd and prime, and the last " in both its two senses (a) of not being measured by any (other) number (<as firj fiiTpdadai dpiBfUii) and (£) of not being obtainable by adding numbers together " (<uv pA auytceiaBat, e£ aptQfi&v), a unit not being a number. Of these attributes some are present in all other odd numbers as well, while the last [primeness in the second sense] belongs also to the dyad, but in nothing but the triad are they all present'." The fact can be equally well illustrated from geometry. Thus, e.g. into the definition of a square (Eucl. I., Def. 22) there enter the several notions of figure, four-sided, equilateral, and right-angled, each of which covers more than the notion into which all enter as attributes 4 . Secondly, a definition must be expressed in terms of things which are prior to, and better known than, the things defined'. This is 1 Trendelenburg, Elements Logites Aristotikot, % 50, ' Trendelenburg, ErlSuttrungcn tu den Elementcn dtr oristotdischen Logii, 3 ed. p. 107. On construction as proof of existence in ancient geometry cf. H. G- Zeuthen, Dit gcomctrische Construction ctls * Existtns&euieis il in drr antiktn Geometric (in McUhcmatischt Annaien, 4}. Band). * Anal. pest. II. 13, 96 a 33 — b t. ' Trendelenburg, Erlauttrnngtn, p. 108. ' Topics vr. 4, 141116 sqq. ch. IX, $ 7] THE DEFINITIONS 147 clear, since the object of a definition is to give us knowledge of the thing defined, and it is by means of things prior and better known that we acquire fresh knowledge, as in the course of demonstrations. But the terms " prior " and " better known " are, as usual susceptible of two meanings ; they may mean (1 ) absolutely or logically prior and better known, or (2) better known relatively to us. In the absolute sense, or from the standpoint of reason, a point is better known than a line, a line than a plane, and a plane than a solid, as also a unit is better known than number (for the unit is prior to, and the first principle of, any number). Similarly, in the absolute sense, a letter is prior to a syllable. But the case is sometimes different relatively to us ; for example, a solid is more easily realised by the senses than a plane, a plane than a line, and a line than a point. Hence, while it is more scientific to begin with the absolutely prior, it may, perhaps, be permissible, in case the learner is not capable of following the scientific order, to explain things by means of what is more intelligible to him. "Among the definitions framed on this principle are those of the point, the line and the plane ; all these explain what is prior by means of what is posterior, for the point is described as the extremity of a line, the line of a plane, the plane of a solid." But, if it is asserted that such definitions by means of things which are more intelligible relatively only to a particular individual are really definitions, it will follow that there may be many definitions of the same thing, one for each individual for whom a thing is being defined, and even different definitions for one and the same individual at different times, since at first sensible objects are more intelligible, while to a better trained mind they become less so. It follows therefore that a thing should be defined by means of the absolutely prior and not the relatively prior, in order that there may be one sole and immutable definition. This is further enforced by reference to the requirement that a good definition must state the genus and the differentiae, for these are among the things which are, in the absolute sense, better known than, and prior to, the species (twv AttX&s yvwptfMoTipwv teal irpor^pav tov etBovs tffrtv). For to destroy the genus and the differentia .is to destroy the species, so that the former are prior to the species ; they are also better known, for, when the species is known, the genus and the differentia must necessarily be known also, e.g. he who knows "man" must also know "animal" and "land-animal," but it does not follow, when the genus and differentia are known, that the species is known too, and hence the species is Jess known than they are 1 . It may be frankly admitted that the scientific definition will require superior mental powers for its apprehension ; and the extent of its use must be a matter of discretion. So far Aristotle ; and we have here the best possible explanation why Euclid supplemented his definition of a point by the statement in 1. Def, 3 that the extremities of a line are points and his definition of a surface by I. Def. 6 to the effect that the extremities of a surface are lines. The supplementary expla- 1 'Tafia vi. 4, C41 b 35 — 34, 148 INTRODUCTION [ch. ix. } 7 nations do in fact enable us to arrive at a better understanding of the formal definitions of a point and a line respectively, as is well ex- plained by Simson in his note on Def. I, Simson says, namely, that we must consider a solid, that is, a magnitude which has length, breadth and thickness, in order to understand aright the definitions of a point, a line and a surface. Consider, for instance, the boundary common to two solids which are contiguous or the boundary which divides one solid into two contiguous parts; this boundary is a surface. We can prove that it has no thickness by taking away either solid, when it remains the boundary of the other; for, if it had thickness, the thickness must either be a part of one solid or of the other, in which case to take away one or other solid would take away the thickness and therefore the boundary itself: which is impossible. Therefore the boundary or the surface has no thickness. In exactly the same way, regarding a line as the boundary of two contiguous surfaces, we prove that the line has no breadth ; and, lastly, regarding a point as the common boundary or extremity of two lines, we prove that a point has no length, breadth or thickness. Aristotle on unscientific definitions. Aristotle distinguishes three kinds of definition which are un- scientific because founded on what is not prior (jtfy *W -n-poTipav). The first is a definition of a thing by means of its opposite, e,g. of " good " by means of " bad " ; this is wrong because opposites are naturally evolved together, and the knowledge of opposites is not uncommonly regarded as one and the same, so that one of the two opposites cannot be better known than the other. It is true that, in some cases of opposites, it would appear that no other sort of definition is possible: e.g. it would seem impossible to define double apart from the half and, generally, this would be the case with things which in their very nature {/cad* aura) are relative terms (w/109 Tt \eyerai), since one cannot be known without the other, so that in the notion of either the other must be comprised as well 1 . The second kind of definition which is based on what is not prior is that in which there is a complete circle through the unconscious use in the definition itself of the notion to be defined though not of the name*. Trendelenburg illustrates this by two current definitions, (i) that of magnitude as that which can be increased or diminished, which is bad because the positive and negative comparatives " more " and " less " presuppose the notion of the positive " great," (2) the famous Euclidean definition of a straight line as that which " lies evenly with the points on itself" (££ Xaov to*s e<f>' lavTrjii <rij^eiois Keirai), where "lies evenly" can only be understood with the aid of the very notion of a straight line which is to be defined'. The third kind of vicious definition from that which is not prior is the definition of one of two coordinate species by means of its coordinate (dirtiStypiiftivov), e.g. a definition of " odd " as that which exceeds the even by a unit (the second alternative in Eucl. VII. Def. 7) ; for " odd " and " even " are coordinates, being differentiae of 1 Topki vi. 4, 141 a u — 31. * Hid, 141 a 34— b 6. * Trendelenburg, Erlautentngtn, p. iij. ch. IX. J 7] THE DEFINITIONS 149 number'. This third kind is similar to the first. Thus, says Tren- delenburg, it would be wrong to define a square as "a rectangle with equal sides." Aristotle's third requirement. A third general observation of Aristotle which is specially relevant to geometrical definitions is that "to know what a thing is (ri itrTtv) is the same as knowing why it is (foil ri ia-nv)'." " What is an eclipse ? A deprivation of light from the moon through the interposition of the earth. Why does an eclipse take place? Or why is the moon eclipsed ? Because the light fails through the earth obstructing it What is harmony ? A ratio of numbers in high or low pitch. Why does the high-pitched harmonise with the low-pitched? Because the high and the low have a numerical ratio to one another*." " We seek the cause (ri Sto-rt) when we are already in possession of the /act (rd ot(). Sometimes they both become evident at the same time, but at al! events the cause cannot possibly be known [as a cause] before the fact is known*." " It is impossible to know what a thing is if we do not know that it is'" Trendelenburg paraphrases: "The definition of the notion does not fulfil its purpose until it is made genetic. It is the producing cause which first reveals the essence of the thing The nominal definitions of geometry have only a provisional significance and are superseded as soon as they are made genetic by means of construction." E.g. the genetic definition of a parallelogram is evolved from Eucl. I. 3 1 (giving the construction for parallels) and I, 33 about the lines joining corresponding ends of two straight lines parallel and equal in length. Where existence is proved by construction, the cause and the fact appear together 1 . Again, " it is not enough that the defining statement should set forth the fact, as most definitions do; it should also contain and present the cause ; whereas in practice what is stated in the definition is usually no more than a conclusion (o-vfiiripatrfta). For example, what is quadrature ? The construction of an equilateral right-angled figure equal to an oblong. But such a definition expresses merely the conclusion. Whereas, if you say that quadrature is the discovery of a mean proportional, then you state the reason'." This is better under- stood if we compare the statement elsewhere that " the cause is the middle term, and this is what is sought in all cases'," and the illustra- tion of this by the case of the proposition that the angle in a semi- circle is a right angle. Here the middle term which it is sought to establish by means of the figure is that the angle in the semi-circle is equal to the half of two right angles. We have then the syllogism : Whatever is half of two right angles is a right angle ; the angle in a semi-circle is the half of two right angles ; therefore {conclusion) the angle in a semi-circle is a right angle*. As with the demonstration, so 1 Topiavi, +, 141 b 7—10. > Anal. post. II. 1, 90 a 31. ' Anal. post. II. », 90 & 1} — 11. * ibid. u. 8, 93 * 17. * ibid, 93 a 10. ' Trendelenburg, Erlduttrungtn, p. no. * Di amma II, 1, 413 a 13— JO. * Anal. posU II. a, 00 a 6, •■Mttvu, 94 a 18. ISO INTRODUCTION [ch. ix. §7 it should be with the definition, A definition which is to show the genesis of the thing defined should contain the middle term or cause ; otherwise it is a mere statement of a conclusion. Consider, for instance, the definition of " quadrature " as " making a square equal in area to a rectangle with unequal sides," This gives no hint as to whether a solution of the problem is possible or how it is solved : but, if you add that to find the mean proportional between two given straight lines gives another straight line such that the square on it is equal to the rectangle contained by the first two straight lines, you supply the necessary middle term or cause 1 . Technical term a not defined by Euclid. It will be observed that what is here defined, " quadrature " or " squaring " (TeTpaiyawio-^oc), is not a geometrical figure, or an attribute of such a figure or a part of a figure, but a technical term used to describe a certain problem. Euclid does not define such things ; but the fact that Aristotle alludes to this particular definition as well as to definitions of deflection (tce/ckd-a-Oat) and of verging {vevew) seems to show that earlier text-books included among definitions explanations of a number of technical terms, and that Euclid deliberately omitted these explanations from his Elements as surplusage. Later the tendency was again in the opposite direction, as we see from the much expanded Definitions of Heron, which, for example, actually include a definition of a deflected line (KettXaajUw) ypa/i,^)'. Euclid uses the passive of icTJiv occasionally*, but evidently considered it unnecessary to explain such terms, which had come to bear a recognised meaning. The mention too by Aristotle of a definition of verging (vcveiv) suggests that the problems indicated by this term were not excluded from elementary text-books before Euclid. The type of problem (vev<ri<t) was that of placing a straight line across two lines, e.g. two straight lines, or a straight line and a circle, so that it shall verge to a given point (i.e. pass through -it if produced) and at the same time the intercept on it made by the two given lines shall be of given length. 1 Other passages in Aristotle may be quoted to the tike effect: e.g.- Anal. pest. I. i, 71 b 9 " We consider that we know a particular thing in the absolute sense, as distinct from the sophistical and incidental sense, when we consider that we know the cause on account of which the thing is, in the sense of knowing that it is the cause of that thing and that it cannot be otherwise," ibid. I. 15, 79 a 1 " For here to know the fact is the function of those who are concerned with sensible things, to know the cause [s the function of the mathe* matician ; it is he who possesses the proofs of the causes, and often he does not know the fact." In view of such passages it is difficult to see how Proclus came to write (p. 101, i i) that Aristotle was the originator ( ' Apiaror&ovt Kardp^curroi) of the idea of Ampbtnomus and others that geometry does not investigate the cause and the why (ri 9i& ri}. To this Gerninuj replied that the investigation of the cause does, on the contrary, appear in geometry. " For how can it be maintained that it is not the business of the geometer to inquire for what reason, on the one hand, an infinite number of equilateral polygons are inscribed in a circle, but, on the other hand, it is not possible to inscribe in a sphere an infinite number of polyhedral figures, equilateral, equiangular, and made up of similar plane figures ? Whose business is it to ask this question and find the answer to it if it is not that of the geometer? Now when geometers reason per impoisibilt they are content to discover the property, bat when they argue by direct proof, if such proof be only partial {irl liipovt), this does not suffice for showing the cause ; if however it is general and applies to all like cases, the why (ri fed ri) is at once and concurrently made evident." 1 Heron, Def. 11 (vol.* iv, Heib. pp. 11-14). * e.g. in lit. jo and in Data 89, ch. ix. 57] THE DEFINITIONS 151 In genera!, the use of conies is required for the theoretical solution of these problems, or a mechanical contrivance for their practical solution 1 . Zeuthen, following Oppermann, gives reasons for supposing, not only that mechanical constructions were practically used by the older Greek geometers for solving these problems, but that they were theoretically recognised as a permissible means of solution when the solution could not be effected by means of the straight line and circle, and that it was only in later times that it was considered necessary to use conies in every case where that was possible*. Heiberg' suggests that the allusion of Aristotle to vtvo-em perhaps confirms this sup- position, as Aristotle nowhere shows the slightest acquaintance with conies. I doubt whether this is a safe inference, since the problems of this type included in the elementary text-books might easily have been limited to those which could be solved by " plane " methods (i.e. by means of the straight line and circle). We know, e.g., from Pappus that Apollonius wrote two Books on plane rewetf*. But one thing is certain, namely that Euclid deliberately excluded this class of problem, doubtless as not being essential in a book of Elements. Definitions not afterwards used. I-astly, Euclid has definitions of some terms which he never after- wards uses, e.g. oblong (eTepo/w^s), rhombus, rhomboid. The "oblong" occurs in Aristotle ; and it is certain that all these definitions are survivals from earlier books of Elements. 1 Cf. the chapter on *c6aen in The Works of Archimedes, pp. c — exxii. * Zeutherii Die Lekrt von dm Kegtlsihnittcn im Alttrtum, ch. is, p. 16a. * Heiberg, Matkemaiisches tu A rh tot cits, p. 16. 1 Pappiu VH. pp. 670—1. BOOK I. DEFINITIONS. i. A point is that which has no part. 2. A line is bread thless length, 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The extremities of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself. 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 9. And when the lines containing the angle are straight, the angle is called rectilineal. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. n. An obtuse angle is an angle greater than a right angle. 12. An acute angle is an angle less than a right angle. 13. A boundary is that which is an extremity of any- thing. 14. A figure is that which is contained by any boundary or boundaries. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another ; 154 BOOK I [i. def. 16— post. 4 16. And the point is called the centre of the circle. 17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. 18. A semicircle is the figure contained by the diameter and the circumference cut off by it And the centre of the semicircle is the same as that of the circle. 19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multi- lateral those contained by more than four straight lines. 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal. 21. Further, of trilateral figures, a right-angled tri- angle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute - angled triangle that which has its three angles acute. 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled ; an oblong that which is right-angled but not equilateral ; a rhombus that which is equilateral but not right-angled ; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia. 23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. POSTULATES. Let the following be postulated : 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. i. post. 5— c. n. s] DEFINITIONS ETC. 155 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. COMMON NOTIONS. 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. [7] 4> Things which coincide with one another are equal to one another. [8] 5. The whole is greater than the part Definition i. Si)/i ttov ia-Ttv, uv [icpiK oi$iy. A point is that which has no pari. An exactly parallel use of fiepos (itrrt) in the singular is found in Aristotle, Metaph. 1035 b 32 fitpos /11F ow tort (tat tqv mSovi, literally "There is a part even of the form "; Borsitz translates as if the plural were used, "Theile giebt es," and the meaning is simply "even the form is divisible (into parts)." Accordingly it would be quite justifiable to translate in this case "A point is that which is indivisible into parts." Martianus Capella (5th c. a.d.) alone or almost alone translated differently, "Punctum est cuius pars nihil est," "a point is that a part of which is netting." Notwithstanding that Max Simon (Euclid vnd die sechs planimetrischen Sucker, 1 901) has adopted this translation (on grounds which I shall presently mention), I cannot think that it gives any sense. If a part of a point is nothing, Euclid might as well have said that a point is itself "nothing," which of course he does not do. Pre -Euclidean definitions. It would appear that this was not the definition given in earlier text- books; for Aristotle (Topics vi. 4, 141 b 20), in speaking of "the definitions" of point, line, and surface, says that they alt define the prior by means of the posterior, a point as an extremity of a line, a line of a surface, and a surface of a solid The first definition of a point of which we hear is that given by the Pythagoreans (cf. Proclus, p. 95, 21), who defined it as a "monad having position" or "with position added" (m°vo; irpoo-kafjowra 9i<rw). It is frequently used by Aristotle, either in this exact form (cf. De anima 1. 4, 409 a 6) or its equivalent: e.g. in Metaph. 1016 b 24 he says that that which is indivisible every way in respect of magnitude and qu& magnitude but has not position is a monad, while that which is similarly indivisible and has position is a point. Plato appears to have objected to this definition. Aristotle says (Metaph. IS* BOOK I [i. dkf. i 992 a 20) that he objected "to this genus [that of points] as being a geometrical fiction (ytinptTpiKav Soyita), and called a point the beginning of a line {&px*l ypaft/),^), while again he frequently spoke of * indivisible lines. 1 " To which Aristotle replies that even " indivisible lines " must have extremities, so that the same argument which proves the existence of lines can be used to prove that points exist It would appear therefore that, when Aristotle objects to the definition of a point as the extremity of a line (wipe.? ypap.p.ys) as un- scientific {Topics vi. 4, 141 b 21), he is aiming at Plato. Heiberg conjectures (Mathematisthes m Aristoteks, p. 8) that it was due to Plato's influence that the word for "point" generally used by Aristotle {<my/iij) was replaced by <n)iitiov (the regular term used by Euclid, Archimedes and later writers), the latter term (-nota, a conventional mark) probably being considered more suitable than trrtypy (a. puncture) which might appear to claim greater reality for a point Aristotle's conception of a point as that which is indivisible and has position is further illustrated by such observations as that a point is not a body (l)e caeh 11. 13, 196 a 17) and has no weight (ibid. 111. 1, 299 a 30); again, we can make no distinction between a point and the place (jorm) where it is (Physics iv. 1, 209 a n). He finds the usual difficulty in accounting for the transition from the indivisible, or infinitely small, to the finite or divisible magnitude. A point being indivisible, no accumulation of points, however far it may be carried, can give us anything divisible, whereas of course a line is a divisible magnitude. Hence he holds that points cannot make up anything continuous like a line, point cannot be continuous with point (06 yap l<rrar t^ofitvov tnjpMov trtjfitiov y ariy firj tniy/ofi, De gen. el corr. I. 2, 317 a ro), and a line is not made up of points (ou ^vy«€i™t in emy/uly, Physics iv, 8, a 1 5 b 19). A point, he says, is like the now in time: now is indivisible and is not a. part of time, it b only the beginning or end, or a division, of time, and similarly a point may be an extremity, beginning or division of a line, but is not part of it or of magnitude (cf. De eaelo m. 1, 300 a 14, Physics IV. n, 220 a 1 — 21, vi. 1, 231 b 6 sqq.). It is only by motion that a point can generate a line {De anima 1. 4, 409 a 4) and thus be the origin of magnitude. Other ancient definitions. According to an-Nairizi (ed. Curtee, p. 3) one "Herundes" (not so far identified) defined a point as " the indivisible beginning of all magnitudes," and Position ius as "an extremity which has no dimension, or an extremity of a line." Criticisms by commentators. Euclid's definition itself is of course practically the same as that which Aristotle's frequent allusions show to have been then current, except that it omits to say that the point must have position. Is it then sufficient, seeing that there are other things which are without parts or indivisible, e.g. the now in time, and the unit in number f Proclus answers {p. 93, 18) that the point is the only thing in the subject-matter of geometry that is indivisible. Relatively therefore to the particular science the definition is sufficient Secondly, the definition has been over and over again criticised because it is purely negative. Proclus' answer to this is (p. 94, 10) that negative descriptions are appropriate to first principles, and he quotes Pa mien ides as having described his first and last cause by means of negations merely. Aristotle too admits that it may sometimes be necessary for one framing a definition to use negations, e.g. in defining privative terms such as "blind"; and he seems to accept as proper I okf. i] NOTE ON DEFINITION r 157 the negative element in the definition of a point, since he says (De anima 111,6, 430 b 20) that "the point and every division [e.g. in a length or in a period of time], and that which is indivisible in this sense, ts exhibited as privation Simplicius (quoted by an-Nairizi) says that " a point is the beginning of magnitudes and that from which they grow ; it is also the only thing which, having position, is not divisible." He, like Aristotle, adds that it is by its motion that a point can generate a magnitude : the particular magnitude can only be "of one dimension," viz. a line, since the point does not "spread itself" (dimittat), Simplicius further observes that Euclid defined a point negatively because it was arrived at by detaching surface from body, line from surface, and finally point from line. "Since then body has three dimensions it follows that a point [arrived at after successively eliminating all three dimensions] has none of the dimensions, and has no part," This of course reappears in modern treatises (cf. Rausenberger, Eiementar'gtomctrit des Punktes, der Geraden und der Ebene, 1887, p. 7). An-Nairizi adds an interesting observation. " If any one seeks to know the essence of a point, a thing more simple than a line, let him, in the sensible world, think of the centre of the universe and the poles." But there is nothing new under the sun : the same idea is mentioned, in an Aristotelian treatise, in controverting those who imagine that the poles have some influence in the motion of the sphere, "when the poles have no magnitude but are extremities and points " (De motu animalium 3, 699a 11). Modern views. In the new geometry represented by the excellent treatises which start from new systems of postulates or axioms, the result of the profound study of the fundamental principles of geometry during recent years (I need only mention the names of Pasch, Veronese, Enriques and Hilbert), points come before lines, but the vain effort to define them a priori is not made ; instead of this, the nearest material things in nature are mentioned as illustrations, with the remark that it is from them that we can get the abstract idea. Cf. the full statement as regards the notion of a point in Weber and Wellstein, Encyehpddie dtr eiementaren Mathematik, 11., 1905, p. 9. "This notion is evolved from the notion of tbe real or supposed material point by the process of limits, i.e. by an act of the mind which sets a term to a series of presen- tations in itself unlimited. Suppose a grain of sand or a mote in a sunbeam, which continually becomes smaller and smaller. In this way vanishes more and more the possibility of determining still smaller atoms in the grain of sand, and there is evolved, so we say, with growing certainty, the presentation of the point as a definite position in space which is one and is incapable of further division. But this view is untenable ; we have, it is true, some idea how the grain of sand gets smaller and smaller, but only so long as it remains just visible; after that we are completely in the dark, and we cannot see or imagine the further diminution. That this procedure comes to an end is unthinkable ; that nevertheless there exists a term beyond which it cannot go, we must believe or postulate without ever reaching it . . . It is a pure act of will, not of the understanding." Max Simon observes similarly {Euclid, p. 85) "The notion 'point' belongs to the limit-notions (Grenzbegriffe^, the necessary conclusions of continued, and in themselves unlimited, senes of presentations." He adds, "The point is the limit of localisation; if this is more and more energetically continued, it leads to the limit-notion 'point,' 158 BOOK I [i. deff, r, 2 better 'position,' which at the same time involves a change of notion. Content of space vanishes, relative position remains. 'Point' then, according to our interpretation of Euclid, is the extremest limit of that which we can still think of (not observe) as a spatial presentation, and if we go further than that, not only does extension cease but even relative place, and in this sense the 'part' is nothing." I confess I think that even the meaning which Simon intends to convey is better expressed by "it has no part" than by "the part is nothing," since to take a "part" of a thing in Euclid's sense of the result of a simple division, corresponding to an arithmetical fraction, would not be to change the notion from that of the thing divided to an entirely different one. Definition 2. I 'pafip.!} St nrjiu>% a'wXaT«. A line is breadthless length. This definition may safely be attributed to the Platonic School, if not to Plato himself. Aristotle (Topics vi. 6, 143 b 11) speaks of it as open to objection because it "divides the genus by negation," length being necessarily either breadthless or possessed of breadth ; it would seem however that the objection was only taken in order to score a point against the Platonists, since he says (ibid. 143 b 29) that the argument is "of service only against those who assert that the genus [sc. length] is one numerically, that ts, those who assume ideas," e.g. the idea of length (amo u^*o$) which they regard as a genus : for if the genus, being one and self-existent, could be divided into two species, one of which asserts what the other denies, it would be self- contradictory (WaiU), Proclus (pp. 96, 21—97, 3) observes that, whereas the definition of a point is merely negative, the line introduces the first "dimension," and so its definition is to this extent positive, while it has also a negative element which denies to it the other " dimensions " (Suur-rdtrtii). The negation of both breadth and depth is involved in the single expression "breadthless" (atrXarti), since everything that is without breadth is also destitute of depth, though the converse is of course not tnie. Alternative definitions. The alternative definition alluded to by Proclus, fiiyiOtn iip' ty Suumjov " magnitude in one dimension " or, better perhaps, " magnitude extended one way " (since Suuttoitk as used with reference to line, surface and solid scarcely corresponds to our use of " dimension " when we speak of "one," " two," or " three dimensions "), is attributed by an-Nairlzs to " Heromides," who must presumably be the same as " Herundes," to whom he attributes a certain definition of a point. It appears however in substance in Aristotle, though Aristotle does not use the adjective Sunmirw, nor does he apparently use 8«wrrao-« except of body as having three " dimensions " or " having dimension (or extension J a// ways (vavrg)," the "dimensions" being in his view (1) up and down, (2) before and behind, and (3) right and left, and " up " being the principle or beginning of length, " right " of breadth, and " before " of depth (De cat So 11. 2, 284 b 24). A line is, according to Aristotle, a magnitude " divisible in one way only " (jtattajrg iuLifttrov), in contrast to a magnitude divisible in two ways (&XB ZuxtpiTOv), or a surface, and a magnitude divisible "in all or in three ways" (iramg koX rpixfi jtnipcToV), or a body (Metaph, 1016 b 25 — 27); or it is a magnitude "continuous one way (or in one direction)," as compared with magnitudes continuous tovo ways or three ways, i. def. 2] NOTES ON DEFINITIONS i, 2 159 which curiously enough he describes as " breadth " and " depth " respectively (jtiytdos Zi to p.iv lift tv uvvi^h firjitm, to 8' hrl Suo tAiitos, to 8' twi rpia (3a8o<;, Metaph. 1020 a 11), though he immediately adds that " length " means a line, " breadth " a surface, and " depth " a body. Proclus gives another alternative definition as "flux of a point " (/5ucr« tnjfMuiv), i.e. the path of a point when moved. This idea is also alluded to in Aristode (De anima 1. 4, 409 a 4 above quoted) : " they say that a line by its motion produces a surface, and a point by its motion a line." "This definition," says Proclus (p. 97, 8 — r3), "is a perfect one as showing the essence of the line : he who called it the flux of a point seems to define it from its genetic cause, and it is not every line that he sets before us, but only the immaterial line , for it is this that is produced by the point, which, though itself indivisible, is the cause of the existence of things divisible." Proclus (p. r 00, 5 — 19) adds the useful remark, which, he says, was current in the school of Apollonius, that we have the notion of a line when we ask for the length of a road or a wall measured merely as length ; for in that case we mean something irrespective of breadth, viz. distance in one *' dimension." Further we can obtain sensible perception of a line if we look at the division between the light and the dark when a shadow is thrown on the earth or the moon ; for clearly the division is without breadth, but has length. Species of "lines." After defining the " line " Euclid only mentions one species of line, the straight line, although of course another species appears in the definition of a circle later. He doubtless omitted all classification of lines as unnecessary for his purpose, whereas, for example, Heron follows up his definition of a line by a division of lines into (1) those which are " straight " and {2} those which are not, and a further division of the latter into (a) " circular circumferences," {&) "spiral-shaped" (iiutottScw) lines and (r) "curved" (xajim-vAai) lines generally, and then explains the four terms. Aristotle tells us {Metaph. 986 a 25) that the Pythagoreans distinguished straight (iJW) and curved (ku/u t',W), and this distinction appears in Plato (cf. Republic x. 602 c) and in Aristotle (cf. " to a line belong the attributes straight or curved," Anal. post. 1. 4, 73 b 19; "as in mathematics it is useful to know what is meant by the terms straight and curved," De anima I. 1, 402 b 19). But from the class of " curved " lines Plato and Aristotle separate off the urtpi^ipijs or " circular " as a distinct species often similarly contrasted with straight. Aristotle seems to recognise broken lines forming an angle as one line : thus "a line, if it be bent (xosafi- pin}), but yet continuous, is called one" (Me tap A. 1 01 6 a 2); "the straight line is more one than the bent line" (Hid. 1016 a 12). Cf. Heron, Def. 12, "A broken line (jeckW/u'vij y pappy) so-called is a line which, when produced, does not meet itself." When Proclus says that both Plato and Aristotle divided lines into those which are "straight," "circular" (wtpujxpife) or "a mixture of the two," adding, as regards Plato, that he included in the last of these classes " those which are called helicoidal among plane (curves) and (curves) formed about solids, and such species of curved lines as arise from sections of solids " (p. 104, 1 — 5), he appears to be not quite exact. The reference as regards Plato seems to be to Parmenides 145 B: "At that rate it would seem that the one must have shape, either straight or round (arpoyyvKov) or some combination of the two"; but this scarcely amounts to a formal classification of lines. As regards i6o BOOK I [l. DEF. 2 Aristotle, Proclus seems to have in mind the passage (De each i. 2, 268 b 17) where it is stated that " all motion in space, which we call translation ($apd), is (in) a straight line, a circle, or a combination of the two ; for the first two ate the only simple (motions)." For completeness it is desirable to add the substance of Proclus' account of the classification of lines, for which he quotes Geminus as his authority. Geminus* first classification of lines. This begins {p. in, 1 — 9) with a division of lines into composite [avtSrrtK) and incomposite (io-u'ec'rrov). The only illustration given of the composite class is the "broken line which forms an angle" (9 KftXavjiivy) ™1 ywiW ttoumaa) ; the subdivision of the incomposite class then follows (in the text as it stands the word " composite " is clearly an error for " incomposite "). The subdivisions of the incomposite class are repeated in a later passage (pp. 176, 27 — 177, 23) with some additional details. The following diagram reproduces the effect of both versions as far as possible (all the illustrations mentioned by Proclus being shown in brackets). lines composite (broken line forming an angle) incomposite forming a figure GXytHiTtrroiQvaa.i or determinate (circle, ellipse, eissoid) not forming a figure or indeterminate d6pt&Toi and extending without limit i-r Arttpop iKfia.W&fin'a.t (straight line, parabola, hyperbola, conchoid) The additional details in the second version, which cannot easily be shown in the diagram, are as follows : (1) Of the lines which extend without limit, some do not form a figure at all (viz. the straight line, the parabola and the hyperbola); but some first "come together and form a figure" (i.e. have a loop), "and, for the rest, extend without limit " (p. 177, 8). As the only other curve, besides the parabola and the hyperbola, which has been mentioned as proceeding to infinity is the conchoid (of Nicomedes), we can hardly avoid the conclusion of Tannery 1 that the curve which has a loop and then proceeds to infinity is a variety of the conchoid itself. As is 1 Notes. pour thistoire des ligncs et surf aits tourbes dans Fatttiquiii in Bulletin des sriemts mathim, ct astronam. 1 ser, vm. (1884), pp. 108—0 (Minmires seientifiaues, ri. p. 13). I. def. 2] NOTE ON DEFINITION 2 161 well known, the ordinary conchoid (which was used both for doubling the cube and for trisecting the angle) is obtained in this way. Suppose any number of rays passing through a fixed point (the pole) and intersecting a fixed straight line ; and suppose that points are taken on the rays, beyond the fixed straight line, such that the portions of the rays intercepted between the fixed straight line and the point are equal to a constant distance (hiacrnj/ia), the locus of the points is a conchoid which has the fixed straight line for asymptote. If the "distance" a is measured from the intersection of the ray with the given straight line, not in the direction away from the pole, but towards the pole, we obtain three other curves according as a is less than, equal to, or greater than b, the distance of the pole from the fixed straight line, which is an asymptote in each case. The case in which a ■■■■ l> gives a curve which forms a loop and then proceeds to infinity in the way Proclus describes. Now we know both from Eutocius {Csmm. on Archimedes, ed. Heiberg, in. p. 98) and Proclus (p. 272, 3 — 7) that Nicomedes wrote on conchoidr (in the plural), and Pappus <iv. p. 244, 18) says that besides the "first" (used as above stated) there were "the second, the third and the fourth which are useful for other theorems." (2) Proclus next observes (p. 177, 9) that, of the lines which extend without limit, some are "asymptotic" (an^ummK^ namely "those which never meet, however they are produced," and some are " symptotic" namely " those which will meet sometime " ; and, of the " asymptotic " class, some are in one plane, and others not. Lastly, of the "asymptotic" lines in one plane, some preserve always the same distance from one another, while others continually " lessen the distance, like the hyperbola with reference to the straight line, and the conchoid with reference to the straight line." Geminus' second classification. This (from Proclus, pp. in, 9 — 20 and 112, 16 — 18) can be shown in a diagram thus : Incomposite lines itiimBntx. ypafiftat ' ' ' ? sirnple, drX$ mixed, part making a figure indeterminate tfXyW *oiou<ra A6p4^TCt (e.g. circle) (straight line} lines in planes lines on solids at ir Toft rrtpttiit r; 1 ' 1 line meeting itself extending without limit 4 i* afr-tf tfvfi-Ktnwaix rr i-r Airtipar JKpaWofUnf (e.g. cissoid) lines formed by uctiom tines round solids al Mri rii rtyjdi al rtjA ri mpti. (e.g. conic sections, spirit curves) (e.g. kdix about 1 sphere or about a cone) 1 ; L - — 1 tumotomtrii Hot homfftomertc {cylindrical helix) (all othera) Notes on classes of "lines" and on particular curves. We will now add the most interesting notes found in Proclus with reference to the above classifications or the particular curves mentioned. 16a BOOK I [i. dec. J i. Homoeomeric lines. By this term (ofUHOfLtpcU) are meant lines which are alike in all parts, so that in any one such curve any part can be made to coincide with any other part. Proclus observes that these lines are only three in number, two being "simple" and in a plane {the straight line and the circle), and the third " mixed," {subsisting) " about a solid," namely the cylindrical helix. The latter curve was also called the coehlias or eochlion, and its homoeomeric property was proved by Apollonius in his work n-cpt tov ko^Xwu (Proclus, p. i«5i 5). The fact that there are only three homoeomeric lines was proved by Geminus, "who proved, as a preliminary proposition, that, if from a point (ami rav trtjuuuv, but on p. 251, 4 &$' ' v ° s <7ij/iei'ou) two straight lines be drawn to a homoeomeric line making equal angles with it, the straight lines are equal" {pp. 112, 1—113, 3i cf - P- *5'> 3 ~ l 9)- 2. Mixed lines. It might be supposed, says Proclus (p. 105, n), that the cylindrical helix, being homoeomerie, like the straight line and the circle, must like them be simple. He replies that it is not simple, but mixed, because it is generated by two unlike motions. Two like motions, said Geminus, e.g. two motions at the same speed in the directions of two adjoining sides of a square, produce a simple line, namely a straight line (the diagonal) ; and again, if a straight line moves with its extremities upon the two sides of a right angle respectively, this same motion gives a simple curve (a circle) for the locus of the middle point of the straight line, and a mixed curve (an ellipse) for the locus of any Other point on it (p. 106, 3—15). Geminus also explained that the term " mixed," as applied to curves, and as applied to surfaces, respectively, is used in different senses. As applied to curves, "mixing" neither means simple "putting together" (<rw0«7«) nor " blending" (upturn). Thus the helix (or spiral) is a " mixed " line, but (1) it is not " mixed " in the sense of " putting together," as it would be if, say, part of it were straight and part circular, and (2) it is not mixed in the sense of " blending," because, if it is cut in any way, it does not present the appearance of any simple lines {of which it might be supposed to be compounded, as it were). The " mixing " in the case of lines is rather that in which the con- stituents are destroyed so far as their own character is concerned, and are replaced, as it were, by a chemical combination (feme h avrjj trwi^apfiira to ixpa, koI <rvyic€)(y)i.cra.). On the other hand " mixed " surfaces are mixed in the sense of a sort of " blending " [xatci «™ Kpaa-iy). For take a cone gene- rated by a straight line passing through a fixed point and passing always through the circumference of a circle : if you cut this by a plane parallel to that of the circle, you obtain a circular section, and if you cut it by a plane through the vertex, you obtain a triangle, the " mixed " surface of the cone being thus cut into simple lines (pp. 117, 12 — 118, 23). 3. Spiric curves. These curves, classed with conies as being sections of solids, were dis- covered by Perseus, according to an epigram of Perseus' own quoted by Proclus (p. ii2, t), which says that Perseus found "three lines upon {or, perhaps, in addition to) five sections " (rpew ypn/xtta? hA vltrt Totals). Proclus throws some light on these in the following passages : "Of the spiric sections, one is interlaced, resembling the horse-fetter (finroy irflh)) ; another is widened out in the middle and contracts on each I. def. a] NOTE ON DEFINITION 2 163 side (of the middle), a third is elongated and is narrower in the middle, broadening out on each side of it" (p. 11 a, 4 — 8). " This is the case with the spiric surface ; for it is conceived as generated by the revolution of a circle remaining at right angles [to a plane] and turning about a point which is not its centre [in other words, generated by the revo- lution of a circle about a straight line in its plane not passing through the centre]. Hence the spirt takes three forms, for the centre [of rotation] is either on the circumference, or within it, or without it. And if the centre of rotation is on the circumference, we have the continuous spire (vwixn^t if within, the interlaced {iii.wnrk(yu.evi)), and if without, the open (S«x>js). And the spiric sections are three according to these three differences" (p. 119, 8-r 7 ). " When the hippopede, which is one of the spiric curves, forms an angle with itself, this angle also is contained by mixed lines" (p. 1.27, 1 — 3). " Perseus showed for spirics what was their property {a-vinrr^iia) " (P- 356. ")■ Thus the spiric surface was what we call a tore, or (when open) an anchor- ring. Heron (Def. 97) says it was called alternatively spire {uirflpa} or ring (xpuur;); he calls the variety in which "the circle cuts itself," not "interlaced," but " cross ing-ltse If" (iTrakkaTTUwra). Tannery 1 has discussed these passages, as also did Schiaparelli*. It is clear that Prochis' remark that the difference in the three curves which he mentions corresponds to the difference between the three surfaces is a slip, due perhaps to too hurried transcribing from Geminus ; all three arise from plane sections of the open anchor-ring. If r is the radius of the revolving circle, a the distance of its centre from the axis of rotation, d the distance of the plane section (supposed to be parallel to the axis) from the axis, the three curves described in the first extract correspond to the following cases : (1) d=a~r. In this case the curve is the hippopede, of which the lemniscate of Bernoulli is a particular case, namely that in which a = zr. The name hippopede was doubtless adopted for this one of Perseus' curves on the ground of its resemblance to the hippopede of Eudoxus, which seems to have been the curve of intersection of a sphere with a cylinder touching it internally. (2) a + r>d>a. Here the curve is an oval, (3) a > rf> a - r. The curve is now narrowest in the middle. Tannery explains the " three lines upon (in addition to) five sections " thus. He points out that with the open tore there are two other sections corresponding to (4) d= a : transition from (2) to (3). (5) a- r>d> o, in which case the section consists of two symmetrical ovals. He then shows that the sections of the closed or continuous tore, corre- sponding to a = r, give curves corresponding to (2), (3) and (4) only. Instead of (t) and (5) we have only a section consisting of two equal circles touching one another. On the other hand, the third spire (the interlaced variety) gives three new forms, which make a group of three in addition to the first group oifive sections. 1 Poiw fkisteire des lignts et surfaces conrbts dans tantiquiti in Bulletin tits stientes mat him. et astranom. vill. (i88+|, pp. 35— % 7 [Mlnairts stientifiqiw, II. pp. 14— 18). s Du kemocentrischen SpAarcn des EudaxuSi ties Kallippus und dts Aristottles {Adhartd* lungm tur Gesch. der Math. 1. Heft, 187;, pp. 14Q — <S*)> i6 4 BOOK J [l. DF-F. a The difficulty which I see in this interpretation is the fact that, just after " three lines on five sections " are mentioned, Proclus describes three curves which were evidently the most important ; but these three belong to three of the five sections of the open tore, and are not separate from them, 4. The cissoid. This curve is assumed to be the same as that by means of which, according to Eutocius (Comm. on Archimedes, in. p. 66 sqq.), Diodes in his book ir«pi ■Kvpiwv (On burning-glasses) solved the problem of doubling the cube. It is the locus of points which he found by the following construction. Let AC, BD be diameters at right angles in a circle with centre O. Let E, Fbe points on the quadrants BC, BA respectively such that the arcs BE, BE ait equal. Draw EG, FH perpendicular to CA. D Join AE, and let P be its intersection with FH. The cissoid is the locus of all the points P corresponding to different posi- tions of E on the quadrant BC and of F at an equal distance from B along the arc BA. A ts the point on the curve correspond- ing to the position C for the point E, and B the point on the curve corresponding to the position of E in which it coincides with B. It is easy to see that the curve extends in the direction AB beyond B, and that CK drawn perpendicular to CA is an asymptote. It may be regarded also as having a branch AD symmetrical with AB, and) beyond D, approaching KC produced as asymptote. If OA, 0£> are coordinate axes, the equation of the curve is obviously /(« + *) = ("-*)*> where a is the radius of the circle. There is a cusp at A, and it agrees with this that Proclus should say (p. 1 a 6, 34^ that "cissoidal lines converging to one point like the leaves of ivy — for this is the origin of their name — form an angle." He makes the slight correction (p. 1 28, 5) that it is not two farts of a curve, but one curve, which in this case makes an angle. But what is surprising is that Proclus seems to have no idea of the curve passing outside the circle and having an asymptote, for he several times speaks of it as a closed curve (forming a figure and including an area) : cf. p. 152, 7, "the plane (area) cut off by the cissoidal line has one bounding (line), but it has not in it a centre such that all (straight lines drawn to the curve) from it are equal." It would appear as if Proclus regarded the cissoid as formed by the/our symmetrical cissoidal arcs shown in the figure. Even more peculiar is Proclus' view of the 5, "Single-turn Spiral." This is really the spiral of Archimedes traced by a point starting from the fixed extremity of a straight line and moving uniformly along it, while /' .--'' ° H '•'-.J 5 E 7 s~"^ / j !. Dvrr. 2—4] NOTES ON DEFINITIONS 2—4 165 simultaneously the straight line itself moves uniformly in a plane about its fixed extremity. In Archimedes the spiral has of course any number of turns, the straight line making the same number of complete revolutions. Yet I'roclus, while giving the same account of the generation of the spiral (p. 180, 8 — 12), regards the single-turn spiral us actually stopping short at the point reached after one complete revolution of the straight line : " it is necessary to know that extending without limit is not a property of all lines ; for it neither belongs to the circle nor to the cissoid, nor in general to lines which form figures ; nor even to those which do not form figures. For even the single- turn spiral does not extend without limit— -for it is constructed between two points^nor does any of the other lines so generated do so" (p. 187, 19 — 25). It is curious that Pappus (vm. p. n 10 sqq. } uses the same term finvoVrpoi^ns \\ii to denote one turn, not of the spiral, but of the cylindrical helix. Definition 3. Tp*Hprj$ SI Tripara {TTj^Mld. The extremities of a line art points. It being unscientific, as Aristotle said, to define a point as the " extremity of a line " (iripas ypa/iftjjs), thereby explaining the prior by the posterior, Euclid defined a point differently ; then, as it was necessary to connect a point with a line,, he introduced this explanation after the definitions of both had been given. This compromise is no doubt his own idea; the same thing occurs with reference to a surface and a line as its extremity in Def. 6, and with reference to a solid and a surface as its extremity in xt. Def. 2. We miss a statement of the facts, equally requiring to be known, that a " division " (Suupio-ij) of a line, no less than its " beginning " or " end," is a point (this is brought out by Aristotle: cf, Metapk. 1060 b 15), and that the intersection of two lines is also a point. If these additional explanations had been given, Proclus would have been spared the difficulty which he finds in the fact that some of the tines used in Euclid (namelv infinite straight lines on the one hand, and circles on the other) have no " extremities." So also the ellipse, which Proclus calls by the, old name (foptfc (" shield "). In the case of the circle and ellipse we can, he observes (p. 105, 7), take a portion bounded by points, and the definition applies to that portion. His rather far-fetched distinction between two aspects of a circle or ellipse as a line and as a closed figure (thus, while you are describing a circle, you have two extremi- ties at any moment, but they disappear when it is finished) is an unnecessarily elaborate attempt to establish the literal universality of the "definition," which is really no more than an explanation that, if a line has extremities, those extremities are points. Definition 4. Eu#eiu yfnififurj tarty, $rtf ci laov toIs i<f tavriji <r<i(fi(ot* xcirai. A straight line is a line which lies evenly with the points on itself. The only definition of a straight line authenticated as pre-Euclidean is that of Plato, who defined it as " that of which the middle covers the ends " (relatively, that is, to an eye placed at either end and looking along the straight line). It appears in the Parmenidcs 137 k: "suaight is whatever has its middle in front of (i.e. so placed as to obstruct the view of) both its ends " 1 66 BOOK I [i. dkf. 4 (diflii yt ov iv to /iitror ap.tj>oir ratv iaxdrotv btiTrpottOfv p). Aristotle quotes it in equivalent terms {Topics vi. n, 148 b 27), o5 to /h'o-oc &r«rnotr#«I row inpao-ii' ; and, as he does not mention the name of its author, but states it in combina- tion with the definition of a line as the extremity of a surface, we may assume that he used it as being well known. Proclus also quotes the definition as Plato's in almost identical terms, fc to fU<ra row oupow is-tirpoo-fltt (p. 109, ji). This definition is ingenious, but implicitly appeals to the sense of sight and involves the postulate that the line of sight is straight. (Cf. the Aristotelian Problems 31, to, 950 a 39, where the question is why we can better observe straightness in a row, say, of letters with one eye than with two.) As regards the straightness of "visual rays," oijrui, cf. Euclid's own Optics, Ueff. i, *, assumed as hypotheses, in which he first speaks of the " straight lines " drawn from the eye, avoiding the word ttyws, and then says that the figure contained by the visual rays (oif/us) is a cone with its vertex in the eye. As Aristotle mentions no definition of a straight line resembling Euclid's, but gives only Plato's definition and the other explaining it as the " extremity of a surface," the latter being evidently the current definition in contemporary textbooks, we may safely infer that Euclid's definition was a new departure of his own. Proclus on Euclid's definition. Coming now to the interpretation of Euclid's definition, tW«a ypa/1^17 t<mv, ljTis i£ utou ToTi i$' (ttvnji otjiiiIok KctTat, we find any number of slightly different versions, but none that can be described as quite satisfactory ; some authorities, e.g. Savile, have confessed that they could make nothing of it It is natural to appeal to Proclus first ; and we find that he does in fact give an interpretation which at first sight seems plausible. He says {p. 109, 8 sq.) that Euclid "shows by means of this that the straight line alone [of all lines] occupies a distance (mr^iiv $cacn;/ia) equal to that between the points on it. For, as far as one of the points is distant from another, so great is the length (p.fytSm) of the straight line of which they are the extremities ; and this is the meaning of lying i£ «™v to (or with) the points on it " \i( urov being thus, apparently, interpreted as "at" (or "over") "an equal distance"]. "But if you take two points on the circumference (of a circle) or any other line, the distance cut off between them along the line is greater than the interval separating them. And this is the case with every line except the straight line. Hence the ordinary remark, based on a common notion, that those who journey in a straight line only travel the necessary distance, while those who do not go straight travel more than the necessary distance." (Cf. Aristotle, De caclo 1. 4, 271 a r3, "we always call the distance of anything the straight line" drawn to it.) Thus Proclus would interpret somewhat in this way: "a straight line is that which represents extension equal with (the distances separating) the points on it." This explanation seems to be an attempt to graft on to Euclid's definition the assumption (it is a Xunfiavoptvor, not a definition) of Archimedes {On the sphere and cylinder 1. ad init.) that "of all the lines which have the same extremities the straight line is least." For this purpose i£ «row has apparently to be taken as meaning "at an equal distance," and again "lying at an equal distance" as equivalent to "extending over (or representing) an equal distance." This is difficult enough in itself, but is seen to be an impossible interpretation when applied to the similar definition of a plane by Euclid (Def. 7) as a surface "which lies evenly with the straight lines on itself." In that connexion Proclus tries to make the same words «£ Iirou i. def, 4] NOTE ON DEFINITION 4 167 ■ciroi mean "extends over an equal area with." He says namely (p. 117, 2) that, "if two straight lines are set out " on the plane, the plane surface "occupies a space equal to that between the straight lines." But two straight lines do not determine by themselves any space at all j it would be necessary to have a dosed figure with its boundaries in the plane before we could arrive at the equivalent of the other assumption of Archimedes that "of surfaces which have the same extremities, if those extremities are in a plane, the plane is the least [in area]." This seems to be an impossible sense for i£ urov even on the assumption that it means " at an equal distance " in the present definition. The necessity therefore of interpreting i£ Urov similarly in hoth definitions makes it impossible to regard it as referring to distance or length at all. It should be added that Simplicius gave the same explanations as Proclus (an-Nairtzl, p. 5). The language and construction of the definition. Let us now consider the actual wording and grammar of the phrase ^ns i£ Urov toU i<j> tauTtji <rrj(uioK xttTai. As regards the expression if urov we note that Plato and Aristotle (whose use of it seems typical) commonly have it in the sense of "on a footing of equality": cf. oi i( EVou in Plato's Laws 777 d, gig d; Aristotle, Politics 1259 b g «£ ta-uv ttvat (fovkerai tt/v $va-w t "tend to be on an equality in nature," Eth. Nit. vm. 12, 1161 a 8 tVraWa ira>r« i( Urov, " there all are on a footing of equality." Slightly different are the uses in Aristotle, Eth, Nie. x. 8, 1178 a 25 rur /ikr yap iwyimri'w xP tul *<" i£ Urov Itrrta, "both need the necessaries of life to the same extent, Set us say"; Topics ix, 15, 174 a 32 i$ urov iroioWa rije tpiinjatv, "asking the question indifferently" {i.e. without showing any expectation of one answer being given rather than another). The natural meaning would therefore appear to be "evenly placed" (or balanced), "in equal measure," " indifferently" or "without bias" one way or the other. Next, is the dative rots i<f>' io.vrr}t o"i)/u£ok constructed with i( urov or with MttBt? In the first case the phrase must mean "that which he&evenfy with (or in respect to) the points on it," in the second apparently "that which, in (or by) the points on it, lies (or is placed) evenly (or uniformly)." Max Simon takes the first construction to give the sense "die Gerade liegt in gleicher Weise wie ihre Punkte." If the last words mean " in the same way as (or in like manner as) its points," I cannot see that they tell us anything, although Simon attaches to the words the notion of distance (Abstand) like Proclus. The second construction he takes as giving " die Gerade liegt fur (durch) ihre Punkte gleichmassig," " the straight line lies symmetrically for (or through) its points"; or, if k«t<h is taken as the passive of Ti'Siff«, "die Gerade ist durch ihre Funkte gleichmassig gegeben worden," " the straight line is symmetrically determined by its points." He adds that the idea is here direction, and that both direction and distance (as between two different given points simply) would be to Euclid, as later to Bolzano {Betrachtungen iiber einige Gegenstdnde der Elementargeometrie, 1804, quoted by Schotten, Inhalt and Metkode des planimetrischtn Unttrrichts, ». p. 16), primary irreducible notions. While the language is thus seen to be hopelessly obscure, we can safely say that the sort of idea which Euclid wished to express was that of a line which presents the same shape at and relatively to all points on it, without any irregular or unsymmetrical feature distinguishing one part or side of it from another. Any such irregularity could, as Saccheri points out (Engel and Stackel, Die Theorie der Parallellinien von Euklidbis Gauss, 1895, p. 109), be at once made perceptible by keeping the ends fixed and turning the line about 1 68 BOOK I [i. def. 4 them right round; if any two positions were distinguishable, e.g. one being to the left or right relatively to another, " it would not lie in a uniform manner between its points." A conjecture as to its origin and meaning. The question arises, what- was the origin of Euclid's definition, or, how was it suggested to him ? It seems to me that the basis of it was really Plato's definition of a straight line as " that line the middle of which covers the ends." Euclid was a Platonist, and what more natural than that he should have adopted Plato's definition in substance, while regarding it as essential to change the form of words in order to make it independent of any implied appeal to vision, which, as a physical fact, could not properly find a place in a purely geometrical definition ? I believe therefore that Euclid's definition is simply an attempt (albeit unsuccessful, from the nature of the case) to express, in terms to which a geometer could not object as not being part of geometrical subject-matter, the same thing as the Platonic definition. The truth is that Euclid was attempting the impossible. As Pfieiderer says (Scholia to Euclid), "It seems as though the notion of a straight line, owing to its simplicity , cannot be explained by any regular definition which does not introduce words already containing in themselves, by implication, the notion to be defined (such e.g. are direction, equality, uniformity or evenness of position, unswerving course), and as though it were impossible, if a person does not already know what the term straight here means, to teach it to him unless by putting before him in some way a picture or a drawing of it." This is accordingly done in such books as Veronese's Elementi ii geometria (Part I., roo4, p. ro): "A stretched string, e.g. a plummet, a ray of light entering by a small hole into a dark room, are rectilineal objects. Hie image of them gives us the abstract idea of the limited line which is called a rectilineal segment," Other definitions. We will conclude this note with some other famous definitions of a straight line. The following are given by Proclus (p. no, 18 — a 3). i, A line stretched to the utmost, bf anpov Ttrafdrr) ypafi./^. This appears in Heron also, with the words " towards the ends " (iwl ri urifiara) added. (Heron, Def. 4). 2. Part of it cannot be in the assumed plane while part is in one higher up (in fitTiiupoTtpui). This is a. proposition in Euclid (xi. 1). 3. All its parts fit on all {other parts) alike, Tavra avnjs ™ fii/nj wairiv biioiun ty<W*o(f ■. Heron has this too (Def. 4), but instead of " alike " he says ravroitos, "in all ways," which is better as indicating that the applied part may be applied one way or the reverse way, with the same result. 4. That line which, when its ends remain fixed, itself remains fixed, ij t«c irtpdruiv ittvovrmv naX avr? /itvowra. Heron's addition to this, " when it is, as it were, turned round in the same plane " (otur iv ry aw& /n-iW&o orpi^o/tcnf), and his next variation, " and about the same ends having always the same position," show that thf; definition of a straight line as "that which does not change its position when it is turned about its extremities (or any two points in it) as poles " was no original discovery of Leibniz, or Saccheri, or Krafft, or Gauss, but goes back at least to the beginning of the Christian era. Gauss' form of this definition was : " The line in which lie all points that, during the revolution of a body (a part of space) about two fixed paints, maintain their position unchanged is called a straight line." Schotten i. deff. 4, 5J NOTES ON DEFINITIONS 4, 5 169 (I- p. 315) maintains that the notion of a straight tine and its property of being determined by two points are unconsciously assumed in this definition, which is therefore a logical "circle." 5. That line which with one other of tkt same species cannot complete a figure, y fA€TCi njs &po<£$ov? /ita; crj^jxa fty dirorfkowsa. This is an obvious tcrripov-icponpov, since it assumes the notion of a figure. Lastly Leibniz' definition should be mentioned: A straight line is one which divides a plane into two halves identical in all but position. Apart from the fact that this definition introduces the plane, it does not seem to have any advantages over the definition last but one referred to. Legendre uses the Archimedean property of a straight line as the shortest distance between two points. Van Swinden observes (Elcmcnte der Geometric, 1834, p. 4), that to take this as the definition involves assuml/tgthe proposition that any two sides of a triangle are greater than the third and proving that straight lines which have two points in common coincide tnroughout their length (cf. Legendre, Aliments de Giom/trie I. 3, 8). The above definitions all illustrate the observation of Unger {Die Geometric dcs Euilid, 1833) : "Straight is a simple notion, and hence all definitions of tt must fail.... But if the proper idea of a straight line has once been grasped, it will be recognised in all the various definitions usually given of it ; all the definitions must therefore be regarded as explanations, and among them that one is the best from which further inferences can immediately be drawn as to the essence of the straight line." Definition 5. Em<ftivtta Si itrroi, $ ft$KM *<" wAaros jaocav Jvtt. A surface it that which has length and breadth only. The word Iru^drua was used by Euclid and later writers to denote surfact in general, while they appropriated the word twiirtSov for plane surface, thus making tiriwtSor a species of the genus i-rt^tdyua. A solitary use of hrt$avtta by Euclid when a plane is meant (xi. Def. 1 1 ) is probably due to the fact that the particular definition came from an earlier textbook. Froclus (p. 116, 17) remarks that the older philosophers, including Plato and Aristotle, used the words brufrdvtia and imn&ov indifferently for any kind of surface. Aristotle does indeed use both words for a surface, with perhaps a tendency to use iVi^oKtio more than iV«r«W for a surface not plane. Cf. Categories 6, 5 a 1 sq., where both words are used in one sentence : " You can find a common boundary at which the parts fit together, a point in the case of a line, and a line in the case of a surface (ort^aVtm); for the parts of the surface (imni&ov) do fit together at some common boundary. Similarly also in the ease of a body you can find a common boundary, a line or a surface (irujuLvftn), at which the parts of the body fit together." Plato however does not use ivupdvtia at all in the sense of surface, but only cjmrtSov for both surface and plane surface. There is reason therefore for doubting the correctness of the notice in Diogenes Laertius, 111. a 4, that Plato "was the first philosopher to name, among extremities, the//o«# surface" («V»VtSo$ rnufmytui). (Tt^ocuo of course means literally the feature of a body which is apparent to the eye (iwi^avrj*), namely the surface. Aristotle tells us (De sensu 3, 439 a 31) that the Pythagoreans called a surface xpoin, which seems to have meant skin as well as colour. Aristotle explains the term with reference to colour (xp&l*") a* a thing inseparable from the extremity (srepai) of a body. i7» BOOK 1 [i. def. s Alternative definitions. The definitions of a surface correspond to those of a line. As in Aristotle a line is a magnitude " (extended) one way, or in one ' dimension ' " (i<j> h), "continuous one way" (i<j> tr trunx^), or "divisible in one way" (^iowiy^ Statpfroi-), so a surface is a magnitude extended or continuous two ways (hi Sw>), or divisible in two ways (Sixi)). As in Euclid a surface has " length and breadth " only, so in Aristotle " breadth " is characteristic of the surface and is once used as synonymous with it (Metaph. tojo a ix), and again "lengths are made up of long and short, surfaces of broad and narrow, and solids (oyi™) of deep and shallow" (Metaph. 1085 a 10). Aristotle mentions the common remark that a line by its motion produces a surface (De anima ]. 4, 409 a 4). He also gives the a posteriori description of a surface as the "extremity of a solid" (Topics vi. 4, [41 b 21}, and as "the section (rofnf) or division {S«up«rii) of a body" (Metaph. 1060 b 14). Proclus remarks (p. 114, jo) that we get a notion of a surface when we measure areas and mark their boundaries in the sense of length and breadth ; and we further get a sort of perception of it by looking at shadows, since these have no depth {for they do not penetrate the earth) but only have length and breadth. Classification of surfaces. Heron gives (Def. 74, p. 50, ed. Heiberg) two alternative divisions or surfaces into two classes, corresponding to Gemirtus' alternative divisions of lines, viz. into (1) incomposite and composite and (2) simple and mixed. (1) Incomposite surfaces are "those which, when produced, fall into (or coalesce with) themselves" (wtim iK0akXo/ityai, airat tiaff iavrwv Wwrowie), i.e. are of continuous curvature, e.g. the sphere. Composite surfaces are "those which, when produced, cut one another." Of composite surfaces, again, some are (a) made up of non-homogeneous (elements) (i$ dvofiotoyow) such as cones, cylinders and hemispheres, others (&) made up of homogeneous (elements), namely the rectilineal (or polyhedral) surfaces. (2) Under the alternative division, simple surfaces are the plane and the spherical surfaces, but no others ; the mixed class includes all other surfaces whatever and is therefore infinite in variety. Heron specially mentions as belonging to the mixed class (a) the surface of cones, cylinders and the like, which are a mixture of plane and circular (fitKTal i£ tTrnrt'Sou Kal irepic^epttas) and (h) spirie surfaces, which are "a mixture of two circumferences " (by which he must mean a mixture of two circular elements, namely the generating circle and its circular motion about an axis in the same plane). Proclus adds the remark that, curiously enough, mixed surfaces may arise by the revolution either of simple curves, e.g. in the case of the spire, or of mixed curves, e.g. the "right-angled conoid" from a parabola, "another conoid" from the hyperbola, the "oblong" (krifinKts, in Archimedes iropa- pa-xts) and " flat " (VwtjrAa-nJ) spheroids from an ellipse according as it revolves about the major or minor axis respectively (pp. 119, 6 — 120, 2). The homoeo- meric surfaces, namely those any part of which will coincide with any other part, are two only (the plane and the spherical surface), not three as in the case of lines (p. no, 7). i. deff. 6, 7] NOTES ON DEFINITIONS 5— 7 i?» Definition 6. 'Eirt^avtias Si -ripara. ypappmi. The extremities of a surface are lines. It being unscientific, as Aristotle says, to define a line as the extremity of a surface, Euclid avoids the error of defining the prior by means of the posterior in this way, and gives a different definition not open to this objection. Then, by way of compromise, and in order to show the connexion between a line and a surface, he adds the equivalent of the definition of a line previously current as an explanation. As in the corresponding Def. 3 above, he omits to add what is made clear by Aristotle (Metapk. 1060 b 15) that a "division" (Siaipco-is) or " section " (to/ii;) of a solid or body is also a surface, or that the common boundary at which two parts of a solid fit together (Categories 6, 5 a a) may be a surface. Proclus discusses how the fact stated in Def. 6 can be said to be true of surfaces like that of the sphere "which is bounded (wtiripairrat), it is true, but not by lines." His explanation (p. 116, 8 — 14) is that, "if we take the surface (of a sphere), so far as it is extended two ways (BtyjJ &a<rr<iTi/), we shall find that it is bounded by lines as to length and breadth ; and if we consider the spherical surface as possessing a form of its own and invested with a fresh quality, we must regard it as having fitted end on to beginning and made the two ends (or extremities) one, being thus one potentially only, and not in actuality." Definition 7, 'Eir(jr«S« brttJMvtta itrrtv, iJtii t£ itrmi rati 1$ iavrijs ivBtiais wttrai. A plane surface is a surface which ties evenly with the straight lines on itself. The Greek follows exactly the definition of a straight line mutatis mutandis , i.e. with Ta«...riS(t'ttit for tok. . .(rryuiW Proclus remarks that, in general, all the definitions of a straight line can be adapted to the plane surface by merely changing the genus. Thus, for instance, a plane surface is " a surface the middle of which covers the ends " (this being the adaptation of Plato's definition of a straight line). Whether Plato actually gave this as the defini- tion of a plane surface or not, I believe that Euclid's definition of a plane surface as lying evenly with the straight lines on itself was intended simply to express the same idea without any implied appeal to vision (just as in the corresponding case of the definition of a straight line). As already noted under Def. 4, Proclus tries to read into Euclid's defini- tion the Archimedean assumption that "of surfaces which have the same extremities, if those extremities are in a plane, the plane is the least." But, as I have stated, his interpretation of the words seems impossible, although it is adopted by Simplicius also (see an-Naiiizi). Ancient alternatives. The other ancient definitions recorded are as follows. 1. The surf act which is stretched to the utmost («V axpav t tropin}) : a definition which Proclus describes as equivalent to Euclid's definition (on Proclus' own view of that definition). Cf. Heron, Def. 9, " (a surface) which is right (and) stretched out " (ip8i) ovtra airortrnjuoTj), words which he adds to Euclid's definition. ija BOOK I [l mf. 7 z. The least surface among all those which have the same extremities. Proclus is here (p. 1 1 7, 9) obviously quoting the Archimedean assumption. 3. A surface all the farts of which have the property of fitting on (each ether) (Heron, Def. 9). 4. A surface such that a straight line fits on all parts of it (Proclus, p. 117, 8), or such that the straight line fits on it all ways, i.e. however placed (Proclus, p. 117, jo). With this should be compared : 5. "(A plane surface is) such that, if a straight line pass through two points on it, the line coincides wholly with it at every spot, all ways," i.e. however placed (one way or the reverse, no matter how), 7c ixtiSiiv &vo trqutuiiv aijnjTai ritdtla, koX okrj aurp Kara irarra roirctv miPTOtfd? i$apfi4£*Tat (Heron, Def. 9). This appears, with the words «<rra mivra Torov iraiToMot omitted, in Theon of Smyrna (p. 112, 5, ed. Hiller), so that it goes back at least as far as the 1st c. a.d. It is of course the same as the definition commonly attributed to Robert Simson, and very widely adopted as a substitute for Euclid's. This same definition appears also in an-Nairlzl (ed. Curtze, p. 10) who, after quoting Simplicius' explanation (on the same lines as Proclus') of the meaning of Euclid's definition, goes on to say that "others defined the plane surface as that in which it is possible to draw a straight tine from any point to any other." pifficuitiea in ordinary definitions. Gauss observed in a letter to Bessel that the definition of a plane surface as a surface such that, if any him points in it be tahcn, the straight line joining them lies wAolfy in the surface (which, for short, we will call "Simson's" definition) contains more than is necessary, in that a plane can be obtained by simply projecting a straight line lying in it from a point outside the line but also lying on the plane ; in fact the definition includes a theorem, or postulate, as well. The same is true of Euclid's definition of a plane as the surface which "lies evenly with (all) the straight lines on itself," because it is sufficient for a definition of a plane if the surface " lies evenly " with those lines only which pass through a fixed point on it and each of the several points of a straight line also lying in it but not passing through the point But from Euclid's point of view it is immaterial whether a definition contains more than the necessary minimum provided that the existence of a thing possessing all the attributes contained in the definition is afterwards proved. This however is not done in regard to the plane. No proposition about the nature of a plane as such appears before Book XI., although its existence is presupposed in all the geometrical Books 1.— iv. and vi. ; nor in Book xi. is there any attempt to prove, _e.g. by construction, the existence of a surface conforming to the definition. The explanation may be that the existence of the plane as defined was deliberately assumed from the beginning like that of points and lines, the existence of which, according to Aristotle, must be assumed as principles unproved, while the existence of everything else must be proved ; and tt may well be that Aristotle would have included plane surfaces with points and lines in this statement had it not been that he generally took his illustrations from plane geometry (excluding solid). But, whatever definition of a plane is taken, the evolution of its essential properties is extraordinarily difficult. Crelle, who wrote an elaborate article Zur Theorie der Ebene (read in the Academie der Wissenschaften in 1834) of which account must be taken in any full history of the subject, observes that, I. dep. 7] NOTE ON DEFINITION 7 173 since the plane is the field, as it were, of almost all the rest of geometry, while a proper conception of it is necessary to enable Eucl. 1. 1 to be understood, it might have been expected that the theory of the plane would have been the subject of at least the same amount of attention as, say, that of parallels. This however was far from being the case, perhaps because the subject of parallels (which, for the rest, presuppose the notion of a plane) is much taster than that of the plane. The nature of the difficulties as regards the plane have also been pointed out recently by Mr Frank land {The First Book of Euclid's Elements, Cambridge, 1905) ; it would appear that, whatever definition is taken, whether the simplest (as containing the minimum necessary to deter- mine a plane) or the more complex, e.g. Simson's, some postulate has to be assumed in addition before the fundamental properties, or the truth of the other definitions, can be established. Crelle notes the same thing as regards Simson's definition, containing mare than is necessary. Suppose a plane in which lies the triangle ABC. Let AD join the vertex A to any point D on EC, and BE the vertex B to any point E on CA. Then, according to the definition, AD lies wholly in the plane of the triangle; so does BE. But, if both AD and BE are to lie wholly in the one plane, AD, BE must intersect, say at F: if they did not, there would be two planes in question, not one. But the fact that the lines intersect and that, say, AD does not pass above or below BE, is by no means self-evident Mr Frankland points out the similar difficulty as regards the simpler definition of a plane as the surface generated by a straight line passing always through a fixed point and always intersecting a fixed straight line. Let OFF, OQQ drawn from intersect the straight line X at P, Q respectively. Let E be any third point on X : then it needs to be proved that OR intersects P'Q in some point, say R'. Without some postulate, however, it is not easy to see how to prove this, or even to prove that P'Q intersects X. Crelle'a essay. Definitions by Fourier, Deahna, Becker, Crelle takes as the standard of a good definition that it shall be, not only as simple as possible, but also the best adapted for deducing, with the aid of the simplest possible principles, further properties belonging to the thing defined. He was much attracted by a very lucid definition, due, he says, to Fourier, according to which a flam is formed by the aggregate of ail. the straight lints which, passing through one point on a straight line in space, are perpendicular to that straight line. (This is really no more than an adaptation from Euclid's proposition Xi. 5, to the effect that, if one of four concurrent straight lines be at right angles to each of the other three, those three are in one plane, which proposition is also used in Aristotle, Meteorohgica m. 3, 373 a 13.) But Crelle confesses that he had not been able to deduce the necessary properties from this and had had to substitute the definition, already mentioned, of a plane as the surface containing, throughout their whole length, all the straight lines passing through a fixed point and also intersecting a straight line in space ; and he only claims to have proved, after a long series of propositions, that the " Fourier "- or " perpendicular "-surface and the plane of the other definition just given are identical, after which the properties of the "Fourier "-surface can be used along with those of the plane. The advantage of the Fourier definition is that it leads easily, by means of the two propositions that 174 BOOK I |l def. 7 triangles are equal in all respects (i) when two sides and the included angle are respectively equal and (j) when all three sides are respectively equal, to the property expressed in Simsou's definition. But Crelle uses to establish these two congruence-theorems a number of propositions about equal angles, supple- mentary angles, right angles, greater and less angles ; and it is difficult to question the soundness of Schotten's criticism that these notions in themselves really presuppose that of a plane. The difficulty due to Fourier's use of the word " perpendicular," if that were all, could no doubt be got over. Thus Deahna in a dissertation (Marburg, 1837) constructed a plane as follows. Presupposing the notions of a straight line and a sphere, he observes that, if a sphere revolve about a diameter, all the points of its surface which move describe closed curves (circles). Each of these circles, during the revolution, moves along itself, and one of them divides the surface of the sphere into two congruent parts. The aggregate then of the lines joining the centre to the points of this circle forms thepfojie. Again, J. K. Becker (Die Elemente der Geometric, 1877) pointed out that the revolution of a right angle about one side of it produces a conical surface which differs from all other conical surfaces generated by the revolution of other angles in the fact that the particular cone coincides with the cone vertically opposite to it : this characteristic might therefore be taken in order to get rid of the use of the right angle. W. Bolyai and Lobachewsky. Very similar to Deahna's equivalent for Fourier's definition is the device of W. Bolyai and Lobachewsky (described by Frischauf, Elementt der absnlulen Geometric, 1876). They worked upon a fundamental idea first suggested, apparently, by Leibniz, Briefly stated, their way of evolving a plane and a straight line was as follows. Conceive an infinite number of pairs of concentric spheres described about two fixed points in space, O, 0, as centres, and with equal radii, gradually increasing : these pairs of equal spherical surfaces intersect respectively in homogeneous curves (circles), and the " InbegrifiT" or aggregate of these curves of intersection forms a plane. If A be a point on one of these circles {k say), suppose points M, M' to start simultaneously from A and to move in opposite directions at the same speed till they meet at B, say ; B then is "opposite" to A, and A, .ff divide the circumference into two equal halves. If the points A, B be held fast and the whole system be turned about them until O takes the place of 0, and of O, the circle k will occupy the same position as before (though turned a different way). Two opposite points, P, Q say, of each of the other circles will remain stationary during the motion as well as A, B : the " InbegrifT" or aggregate of all such points which remain stationary forms a straight line. It is next observed that the plane as defined can be generated by the revolution of the straight line about 0(7, and this suggests the following construction for a plane. Let a circle as one of the curves of intersection of the pairs of spherical surfaces be divided as before into two equal halves &tA,B. Let the arc ADB be similarly bisected at D, and let C be the middle point of AB. This determines a straight line CD which is then defined as "perpendicular" to AB. The revo- lution of CD about AB generates a plane. The property stated in Simson's definition is then proved by means of the congruence-theorems proved in Eucl. 1. 8 and 1. 4. The first is taken as proved, practically by considerations of symmetry and homogeneity. If two spherical surfaces, not necessarily equal, with centres O, intersect, A and its "opposite" point B are taken as L D:r. 7] NOTE ON DEFINITION 7 175 before on the curve of intersection (a circle) and, relatively to 00', the point A is taken to be convertible with B or any other point on the homogeneous curve. The second (that of Eucl. i. 4) is established by simple application. Rausenberger objects to these proofs on the grounds that the first assumes that the two spherical surfaces intersect in one single curve, not in several, and that the second compares angles : a comparison which, be says, is possible only in a plane, so that a plane is really presupposed. Perhaps as regards the particular comparison of angles Rausenberger is hypercritical ; but it is difficult to regard the supposed proof of the theorem of Eucl. 1. 8 as sufficiently rigorous (quite apart from the use of the uniform motion of points for the purpose of bisecting lines). Simson's property is proved from the two congruence-theorems thus. Suppose that AB is " perpendicular " (as defined by Bolyai) to two generators CM, CN of a plane, or suppose CM, CN respectively to make with AB two angles congruent with one another. It is enough to prove that, if P be any point on the straight line MN, then CP, just as much as CM, CN respectively, makes with AB two angles congruent with one another and is therefore a generator. We prove successively the congruence of the following pairs of triangles : ACM, BCM ACN, BCN AMN, BMN AMP, BMP ACP, BCP, whence the angles A CP, BCP are congruent. Other views. Enriques and Amaldi (Ekmenti di geometria, Bologna, 1905), Veronese (in his Elementi) and Hilbert all assume as a postulate the property stated in Simson's definition. But G, Ingrami {Elementi di geometria, Bologna, 1904) proves it tn the course of a remarkable series of closely argued proposition based upon a much less comprehensive postulate. He evolves the theory of the plane from that of a triangle, beginning with a triangle as a mere three-side (trilatero), i.e. a frame, as it were. His postulate relates to the three-side and is to the effect that each " (rectilineal) segment " joining a vertex to a point of the opposite side meets every segment similarly joining each of the other two vertices to the points of the sides opposite to them respectively, and, con- versely, if a point be taken on a segment joining a vertex to a point of the opposite side, and if a straight line be drawn from another vertex to the point on the segment so taken, it will if produced meet the opposite side. A triangle is then defined as the figure formed by the aggregate of all the segments joining the respective vertices of a three-side to points on the opposite sides. After a series of propositions, Ingrami evolves a plane as the figure formed by the " half straight-lines " which project from an internal point of the triangle the points of the perimeter, and then, after two more theorems, proves that a plane is determined by any three of its points which are not in a straight line, and that a straight line which has two points in a plane has all its points in it. The argument by which Bolyai and I>obachewsky evolved the plane is of course equivalent to the definition of a plane as the locus of all points equidistant from two fixed points in space. iy6 BOOK I [i. deff, 7—9 Leibniz in a letter to Giordano defined a plane as thai surface which divides space into turn congruent farts. Adverting to Giordano's criticism that you could conceive of surfaces and lines which divided space or a plane into two congruent parts without being plane or straight respectively, Beez ( liber Euklidischc und Nicht-Euklidische Geometric, 1888) pointed out that what was wanted to complete the definition was the further condition that the two congruent spaces could be slid along each other without the surfaces ceasing to coincide, and claimed priority for his completion of the definition in this way. But the idea of all the parts of a plane fitting exactly on all other parts is ancient, appearing, as we have seen, in Heron, Def. 9. Definitions 8, 9. 8. '£mV«Sos &i yiMivla iariv y iv cwnri&w Svo ypafijjL&v OriTTOfUvtav a\kijktitv jcat i*t] tV iiOua.% Ktifiivtiiv TTfm aAAijXuf ruv ypu^/Hav xAurtt. 9. "Orav &* al Trfptixowai ttjv yutvcW ypaftfXal tv&4itu umtiv, tv&uypaftftos jraA.«T<u )J ymria, 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line, 9. And when the lines containing the angle are straight, the angle is called rectilineal. The phrase " not in a straight line " is strange, seeing that the definition purports to apply to angles formed by curves as well as straight lines. We should rather have expected continuous (tTwtxfc) with one another; and Heron takes this to be the meaning, since he at once adds an explanation as to what is meant by lines not being continuous (oi <rw<x<»)- It looks as though Euclid really intended to define a rectilineal angle, but on second thoughts, as a concession to the then common recognition of curvilineal angles, altered " straight lines " into " lines " and separated the definition into two. I think all our evidence suggests that Euclid's definition of an angle as inclination (*A£«s) was a new departure. The word does not occur in Aristotle ; and we should gather from him that the idea generally associated with an angle in his time was rather deflection or breaking of lines (kKiIit^) : cf. his common use of xtK\a<r6at and other parts of the verb nASf, and also his reference to one bent line forming an angle (tt)i/ ntKapjiivipr vol lyaotrav ftriav 1 Metaph. 1016 a 13) Proclus has a long and elaborate note on this definition, much of which (pp. i2i t 12 — 126, 6) is apparently taken direct from a work by his master Syrian us (6 iJfiiTtpo! m&vjnpimrL Two criticisms contained in the note need occasion no difficulty. One of these asks how, if an angle be an inclination, one inclination can produce two angles. The other (p. 1 28, 2) is to the effect that the definition seems to exclude an angle formed by one and the same curve with itself, e.g. the complete cissoid [at what we call the " cusp "1 or the curve known as the hippopede (horse-fetter) [shaped like a lemniscate]. But such an "angle" as this belongs to higher geometry, which Euclid may well be excused for leaving out of account in any case. Other ancient definitions: Apollonius, Plutarch, Carpus. Proclus' note records other definitions of great interest. Apollonius defined an angle as a contracting of a surface or a solid at one point under a broken line Or surface ((rvmyuyq ift^BWMM 4 <7T«p«o5 vpot Iri oij/uiy viti <t((cAa<r^tv); ypawji y iruf>arflf), where again an angle is supposed to be formed by one broken line or surface. Still more interesting, perhaps, is the definition by " those who say that the first distance under the point (ri •wp&rar i. Dbff. 8, 9] NOTES ON DEFINITIONS 7—9 177 &uiim)iia thro to mffuiw) is the angle. Among these is Plutarch, who insists that Apollonius meant the same thing ; for, he says, there must be some first distance under the breaking (or deflection) of the including lines or surfaces, though, the distance under the point being continuous, it is impossible to obtain the actual first, since every distance is divisible without limit" (**•' ajrttpov). There is some vagueness in the use of the word " distance" (Suwrnj/ia) ; thus it was objected that " if we anyhow separate off the first " (distance being apparently the word understood) " and draw a straight line through it, we get a triangle and not one angle." In spite of the objection, I cannot but see in the idea of Plutarch and the others the germ of a valuable conception in infinitesimals, an attempt (though partial and imperfect) to get at the rate of divergence between the lines at their point of meeting as a measure of the angle between them. A third view of an angle was that of Carpus of Antioch, who said " that the angle was a quantity (too-ov), namely a distance (SuMmf/m) between the lines or surfaces containing it This means that it would be a distance (or divergence) in one sense (i<fi tv Siurrwc), although the angle is not on that account a straight line. For it is not everything extended in one sense (to 1$ tv Biao-raToV) that is a line." This very phrase " extended one way" being held to define a line, it is natural that Carpus' idea should have been described as the greatest possible paradox (wdrrw TrupoSo^orarov). The difficulty seems to have been caused by the want of a different technical term to express a new idea ; for Carpus seems undoubtedly to have been anticipating the more modern idea of an angle as representing divergence rather than distance, and to have meant by <+' %v in one sense (rotationatly) as distinct from one way or in one dimension (linearly). To what category does an angle belong ? There was much debate among philosophers as to the particular category (according to the Aristotelian scheme) in which an angle should be placed ; is it, namely, a quantum (voaov), quale (iroioV) or relation (irpos ti)? I. Those who put it in the category of quantity argued from the fact that a plane angle is divided by a line and a solid angle by a surface. Since, then, it is a surface which is divided by a line, and a solid which is divided by a surface, they felt obliged to conclude that an angle is a surface or a solid, and therefore a magnitude. But homogeneous finite magnitudes, e.g. plane angles, must bear a ratio to one another, or one must be capable of being multiplied until it exceeds the other. This is, however, not the case with a rectilineal angle and the horn-like angle (jiton™*^), by which tatter is meant the "angle" between a circle and a tangent to it, since (Eucl. in. 16) the latter "angle" is less than any rectilineal angle whatever. The objection, it will be observed, assumes that the two sorts of angl<£ are homogeneous. Plutarch and Carpus are classed among those who, in one way or other, placed an angle among magnitudes; and, as above noted, Plutarch claimed Apollonius as a supporter of his view, although the word contraction (of a surface or solid) used by the latter does not in itself suggest magnitude much more than Euclid's inclination. It was this last consideration which doubtless led " Aganis," the " friend " (socius) apparently of Simplicius, to substitute for Apollonius' wording " a quantify which has dimensions and the extremities of which arrive at one point" (an-NairUE, p. 13). 3. Eudemus the Peripatetic, who wrote a whole work on the angle, main- tained that it belonged to the category of quality. Aristotle had given as his fourth variety of quality "figure and the shape subsisting in each thing, and, 178 BOOK I [1. Deff. 8, 9 besides these, straight ness, curvature, and the like " (Categories 8, 10 a 11). He says that each individual thing is spoken of as quale in respect of its form, and he instances a triangle and a square, using them again later on (ibid. 1 1 a 5) to show that it is not all qualities which are susceptible of more and less ; again, in Physics 1. 5, 188 a 25 angle, straight, circular are called kinds of figure. Aristotle would no doubt have regarded deflection (xtukairQai) as belonging to the same category with straightness and curvature (KOf«rvA<mft). At all events, Eudemus took up an angle as having its origin in the breaking or deflection («Aao-«) of lines : deflection, he argued, was quality if straightness was, and that which has its origin in quality is itself quality. Objectors to this view argued thus. If an angle be a quality (n-oionjs) like heat or cold, how can it be bisected, say? It can in fact be divided; and, if things of which divisibility is an essential attribute are varieties of quantum and not qualities, an angle cannot be a quality. Further, the more and the less are the appropriate attributes of quality, not the equal and the unequal ; if therefore an angle were a quality, we should have to say of angles, not that one is greater and another smaller, but that one is more an angle and another less an angle, and that two angles are not unequal but dissimilar (ouo'/ioum). As a matter of fact, we are told by Simplicius, 538, at, on Arist De caelo that those who brought the angle under the category of quale did call equal angles similar angles ; and Aristotle himself speaks of similar angles in this sense in De caelo 296020, 311 b 34. 3. Euclid and all who called an angle an inclination are held by Sy nanus to have classed it as a relation (irpoi ti). Yet Euclid certainty regarded angles as magnitudes ; this is clear both from the earliest propositions dealing specifically with angles, e.g. 1. 9, 13, and also (though in another way) from his describing an angle in the very next definition and always as contained (wtpiixo/Uvr]') by the two lines forming it (Simon, Euclid, p. 28}. Proclus (i.e. in this case Sy nanus) adds that the truth lies between these three views. The angle partakes in fact of all those categories : it needs the quantity involved in magnitude, thereby becoming susceptible of equality, inequality and .the like ; it needs the qualify given it by its form, and lastly the relation subsisting between the lines or planes bounding it. Ancient classification of " angles." An elaborate classification of angles given by Proclus (pp. 126, 7 — 127, 16) may safely be attributed to Ge minus. In order to show it by a diagram it Angles on surfaces in solidt {I* OTcptoil) , I ' ~ — "i on simple surfaces on mixed surfaces (e.g. cones, cylinders) on planes on spherical surfaces 1 ' 1 ' 1 made by simple lines made by "mixed" lines by one of each e. g. the angle made by a (e.g. the angle fanned by an curve, such as the cissoid ellipse and its axis of by and hippppede, with itself) an ellipse and a circle) — ' — " 1 T"* '. ] ine line.circumf. cLrcumf.-circumf. line- eon vex line -concave convex-convex concave-concave mixed, or (e.g. angle of a e.g. korH-likt (dp^fcupw) {a^lraAwl convex-concave semicircle) {Kcflaroeidfa) or "scraper-lilce" (e.g. those of (fwrpotiftii) tunes) I. Dkff, 8, 9] NOTE ON DEFINITIONS 8, 9 179 will be necessary to make a convention about terms. Angles are to be under- stood under each class, " line-circumference " means an angle contained by a straight line and an arc of a circle, " line-convex " an angle contained by a straight line and a circular arc with convexity outwards, and so on in every case. Definitions of angle classified. As for the point, straight line, and plane, so foi the angle, Schotten gives a valuable summary, classification and criticism of the different modern views up to date (Inhalt und Methode des planitnetrisehen Unterrichts, 11., 1893, pp. 94 — 183} ; and for later developments represented by Veronese reference may be made to the third article (by Amaldi) in Questioni riguardanti le matematiche elementari, t, (Bologna, 19 12). With one or two exceptions, says Schotten, the definitions of an angle may be classed in three groups representing generally the following views : 1. The angle is the difference of direction between two straight lines. (With this group may be compared Euclid's definition of an angle as an inclination.) 2. The angle is the quantity or amount (or the measure) of the rotation necessary to bring one of its sides from its own position to thai of the other s:de without its moving out of the plane containing both. 3. The angle is the portion of a plane included between two straight tines in the plane which meet in a point (or two rays issuing from the point). It is remarkable however that nearly all of the text-books which give definitions different from those in group 2 add to them something pointing to a connexion between an angle and rotation : a striking indication that the essential nature of an angle is closely connected with rotation, and that a good definition must take account of that connexion. The definitions in the first group must be admitted to be (autologous, or circular, inasmuch as they really presuppose some conception of an angle. Direction (as between tow given points) may no doubt be regarded as a primary notion ; and it may be defined as "the immediate relation of two points which the ray enables us to realise" (Schotten). But "a direction is no intensive magnitude, and therefore two directions cannot have any quantitative difference " (Biirklen). Nor is direction susceptible of differences such as those between qualities, e.g. colours. Direction is a singular entity : there cannot be different sorts or degrees of direction. If we speak of "a different direction," we use the word equivocally ; what we mean is simply " another " direction. The fact is that these definitions of an angle as a difference of direction unconsciously appeal to something outside the notion of direction altogether, to some conception equivalent to that of the angle itself. Recent Italian views. The second group of definitions are (says Amaldi) based on the idea of the rotation of a straight line or ray in a plane about a point : an idea which, logically formulated, may lead to a convenient method of introducing the angle. But it must be made independent of metric conceptions, or of the conception of congruence, so as to bring out first the notion of an angle, and afterwards the notion of equal angles. The third group of definitions satisfy the condition of not including metric conceptions ; but they do not entirely correspond to our intuitive conception of an angle, to which we attribute the character of an entity in one dimension (as Veronese says) with respect to the ray as element, or an entity in two i8o BOOK I [l Dec 9 dimensions with reference to feints as elements, which may be called an angular sector. The defect is however easily remedied by considering the angle as " the aggregate of the rays issuing from the vertex and comprised in the angular sector," Proceeding to consider the principal methods of arriving at the logical formulation of the first superficial properties of the plant from which a definition of the angle may emerge, Amaldi distinguishes two points of view (l) the genetic, (z) the actual. (i) From the first point of view we consider the cluster of straight lines or rays (the aggregate of all the straight lines in a plane passing through a point, or of all the rays with their extremities in that point) as generated by the movement of a straight line or ray in the plane, about a point. This leads to the post ulation of a closed order, or circular disposition, of the straight lines or rays in a cluster. Next comes the connexion subsisting between the disposition of any two clusters whatever in one, plane, and so on. (2) Starting from the point of view of the actual, we lay the foundation of the definition of an angle in the division of the plane into two parts (half- planes) by the straight line. Next, two straight lines (a, b) in the plane, inter- secting at a point O, divide the plane into four regions which ate called angular sectors (convex) ; and finally the angle (ab) or (6a) may lie defined as the aggregate of the rays issuing from O and belonging to the angular sector which has a and b for sides. Veronese's procedure (in his Elementi) is as follows. He begins with the first properties of the plane introduced by the following definition. The figure given by all the straight lines joining the points of a straight line r to a point P outside it and by the parallel to r through P is called a cluster of straight lines, a cluster of rays, or a plane, according as we consider the element of the figure itself to be the straight line, the ray terminated at P, or a foinl. [It will be observed that this method of producing a plane involves using the parallel to r. This presents no difficulty to Veronese because he has previously defined parallels, without reference to the plane, by means of reflex or opposite figures, with respect to a point O : " two straight lines are called parallel, if one of them contains two points opposite to (or the reflex of) two points of the other with respect to the middle point of a common transversal (of the two lines)." He proves by means of a postulate that the parallel r does belong to the plane Pr, Ingrami avoids the use of the parallel by defining a plane as "the figure formed by the half straight lines which project from an internal point of a triangle (i.e. a point on a line joining any vertex of a three-side to a point of the opposite side) the points of its perimeter," and then defining a cluster of rays as "the aggregate of the half straight lines in a plane starting from a given point of the plane and passing through the points of the perimeter of a triangle containing the point"] Veronese goes on to the definition of an angle. " We call an angle a part of a cluster of rays, bounded by two rays (as the segment is a part of a straight line bounded hy two points). "An angle of the cluster, the bounding rays, of which are opposite, is called a flat angle." Then, after a postulate corresponding to postulates which he lays down for i. Deff. 9-1 a] NOTES ON DEFINITIONS 9—12 181 a rectilineal segment and for a straight line, Veronese proves that ail flat angles are equal to one another. a v e Hence he concludes that "the cluster of rays is a homogeneous linear system in which the element is the ray instead of the point. The cluster being a homogeneous linear system, all the propositions deduced from [Veronese's] Post. 1 for the straight line apply to it, e.g. that relative to the sum and difference of the segments : it is only necessary to substitute the ray for the point, and the angle for the segment." Definitions 10, ti, 12. id. T Orak Si cvdfui hr iv&tlav trra$tura tu? tfj>e£y<> yoivta? urac cIAA^'Xacs TOtp, Ap&i) itcaripa i£r law ywetuie fart, Hal ij l$t<rri}m/ta tv#«a itdOtrm KoXiirai, Ijt TjV ilflilTTTJKfV, 1 1 . A/i/JAeiJi yavia itrriv )J ft«'£o»' ipftjs. 1 2 . '0£tia Si 1} tXA&irmr ipftjs. 10. When a straight line set up on a straight line makes the adjacent angles equal is one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. 11. An obtuse angle is an angle greater than a right angle. 1 2. An acute angle is an angle less than a right angle. i4>t^<s is the regular term for adjacent angles, meaning literally " (next) in order." I do not find the term used in Aristotle of angles, but he explains its meaning in such passages as Physics VI. 1, 131 b 8 : "those things are (next) in order which have nothing of the same kind (tjvyy&h) between them." KaSrrtn, perpendicular, means literally let fall: the full expression is perpen- dicular straight line, as we see from the enunciation of Eucl. 1. 11, and the notion is that of a straight line let fall upon the surface of the earth, s. plumb- line. Proclus (p. 283, 9) tells us that in ancient times the perpendicular was called gnomon-wise (Kara yecJporo), because the gnomon (an upright stick) was set up at right angles to the horizon. The three kinds of angles are among the things which according to the Platonic Socrates {Republic vi. 510 c) the geometer assumes and argues from, declining to give any account of them because they are obvious. Aristotle discusses the priority of the right angle in comparison with the acute (Metaph. 1084 b ;): in one way the right angle is prior, i.e. in being defined (on (Jptirrat) and by its notion (t<^ Aoyijt), in another way the acute is prior, i.e. as being a part, and because the right angle is divided into acute angles ; the acute angle is prior as matter, the right angle in respect of form ; cf. also Metaph. 1035 b 6, "the notion of the right angle is not divided into i8a BOOK I [i. Dt.tr. 12-14 that of an acute angle, but the reverse ; for, when denning an acute angle, you make use of the right angle." Proclus {p. 133, 15) observes that it is by the perpendicular that we measure the heights of figures, and that it is by reference to the right angle that we distinguish the other rectilineal angles, which are otherwise undistinguished the one from the other. The Aristotelian Problems {16, 4, 013 b 36) contain an expression perhaps worth quoting. The question discussed is why things which fall on the ground and rebound make "similar" angles with the surface on both sides of the point of impact ; and it is observed that " the right angle is the limit (opm) of the opposite angles," where however " opposite " seems to mean, not " supplementary " (or acute and obtuse), but the equal angles made with the surface on opposite sides of the perpendicular. Proclus, after his manner, remarks that the statement that an angle less than a right angle is acute is not true without qualification, for ( 1 ) the horn-like angle (between the circumference of a circle and a tangent) is less than a right angle, since it is less tnan an acute angle, but is not an acute angle, while (2) the "angle of a semicircle" (between the arc and a diameter) is also less than a right angle, but is not an acute angle. The existence of the right angle is of course proved in 1. n. Definition 13. *Opos iariv, tivos lati iripas. A boundary is that which is an extremity of anything. Aristotle also uses the words Spot and W/kk as synonymous. Cf. De gen. animal, it. 6, 745 a 6, 9, where in the expression '* limit of magnitude " first one and then the other word is used. Proclus {p, 136, 8) remarks that the word boundary is appropriate to the origin of geometry, which began from the measurement of areas of ground and involved the marking of boundaries. Definition 14. %-^rjyni Itrrt to vtt6 tlvo* ij WW Qpwv Trtptt^ofitvot: A figure is that which is contained Ay any boundary or boundaries. Plato in the Meno observes that roundness (o-rpoyyvkarrfi) or the round is a " figure," and that the straight and many other things are so too j he then inquires what there is common to all of them, in virtue of which we apply the term "figure" to them. His answer is (76 a): "with reference to every figure I say that that in which the solid terminates (tovto, tU S to <rrtp€oy mpaivft) is a figure, or, to put it briefly, a figure is an extremity of a solid." The first observation is similar to Aristotle's in the Physics 1, 5, 188 a 25, where angle, straight, and circular are mentioned as genera of figure. In the Categories 8, 10 a 11, "figure" is placed with straightness and curved ness in the category of quality. Here however " figure " appears to mean shape (ttopifnj) rather than " figure " in our sense. Coming nearer to ''figure' 1 in our sense, Aristotle admits that figure is "a sort of magnitude" {fit anima ill. 1, 425 a i£), and he distinguishes plane figures of two kinds, in language not unlike Euclid's, as contained by straight and circular lines respectively ! "every plane figure is either rectilineal or formed by circular lines (xtpufitp6ypa.ii.por), and the rectilineal figure is contained by several lines, the circular by one line" (De caelo 11. 4, 286 b rj). He is careful to explain that a plane is not a i. Deff. 14-16] NOTES ON DEFINITIONS it— 16 183 figure, nor a figure a plane, but that a plane figure constitutes one notion and is a spaiti of the genus figure {Anal, pott. 11. 3, 00 b 37). Aristotle does not attempt to define figure in general, in fact he says it would be useless : " From this it is clear that there is one definition of soul in the same way as there is one definition of figure; for in the one case there is no figure except the triangle, quadrilateral, and so on, nor is there any soul other than those above mentioned. A definition might be constructed which should apply to all figures but not specially to any particular figure, and similarly with the species of soul referred to. [But such a general definition would serve no purpose.] Hence it is absurd here as elsewhere to seek a general definition which wjll not be properly a definition of anything in existence and will not be applicable to the particular irreducible species before us, to the neglect of the definition which is so applicable" (De anima ir. 3, 414 b 20 — 28}. Comparing Euclid's definition with the above, we observe that by intro- ducing boundary (opm) he at once excludes the straight which Aristotle classed as figure ; he doubtless excluded angle also, as we may judge by (1) Heron's statement that " neither one nor two straight lines can complete a figure," (a) the alternative definition of a straight line as "that which cannot with another line of the same species form a figure," (3) Ge minus' distinction between the line which forms a figure (o-xij/urrosrotowa) and the line which extends indefinitely (tr axttpav J*,9oAA<i/iiifi), which latter term includes a hyperbola and a parabola. Instead of calling figure an extremity as Plato did in the expression "extremity (or limit) of a solid," Euclid describes a figure as that -which has a boundary or boundaries. And lastly, in spite of Aristotle's objection, he does attempt a general definition to cover all kinds of figure, solid and plane. It appears certain therefore that Euclid's definition is entirely his own. Another view of a figure, recalling that of Plato in Mens 76 a, is attributed by Proclus (p. 143, 8) to Posidonius. The latter regarded the figure as the confining extremity or limit {ripat avyiAaor), " separating the notion of figure from quantity (or magnitude) and making it the cause of definition, limitation, and inclusion (rot lipitrSai not TrtrtpatrBtu not t^i irtpioy^). . . Posidonius thus seems to have in view only the boundary placed round from outside, Euclid the whole content, so that Euclid will speak of the circle as a figure in respect of its whole plane (surface) and of its inclusion (from) without, whereas Posidonius (makes it a figure) in respect of its circumference... Posidonius wished to explain the notion of figure as itself limit trig and confining magnitude." Proclus observes that a logical and refining critic might object to Euclid's definition as defining the genus from the species, since that which is enclosed by one boundary and that which is enclosed by several are both species of figure. The best answer to this seems to be supplied by the passage of Aristotle's De anima quoted above. Definitions 15, 16. 15. KtikXof Itrrl irjfijfMi t(b(8w iuro pJas ypa/i/i^s Trtpt(xo/t*fov [ij Kakturat rtpi^tip€LtA t irpo9 %v ££' £fot trrjfitlov rvv crros rov oyrnfjuiTos KttfAtvwv iratrat at irpc*T7ri777ov&ai tvOtlai (Vpos -ryv tqv kvkXov Tr*pt$ipiiav] urtu dXAirfXaif titriv. 16. Kivrpov Si rot kvk\ov to trijfAfioy xaAfirat. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying -within the figure are equal to one another ; 16. And the point is called the centre of the circle. 184 BOOK I [1. Deff. 15, 16 The words ij KaXtirm rtpi^iptia, " which is called the circumference," and Tpot ttjv toS kvkXov wtpuftipttay, " to the circumference of the circle," are bracketed by Heiberg because, although the mss. have them, they are omitted in other ancient sources, viz. Proclus, Taurus, Sextus Empiricus and Boethius, and Heron also omits the second gloss. The recently discovered papyrus Hercuianensis No. 1061 also quotes the definition without the words in question, confirming Heiberg's rejection of them (see Heiberg in Hermes xxxviii., 1903, p. 47), The words were doubtless added in view of the occurrence of the word "circumference" in Deff. 17, 18 immediately following, without any explanation. But no explanation was needed. Though the word irtpi^iptia does not occur in Flato, Aristotle uses it several times (1) in the general sense of con tour without any special mathematical signification, {2) mathematically, with reference to the rainbow and the circumference, as well as an arc, of a circle. Hence Euclid was perfectly justified in employing the word in Deff. 17, 18 and elsewhere, but leaving it undefined as being a word universally understood and not involving in itself any mathematical conception. It may be added that an-Nairizi had not the bracketed words in his text ; for he comments on and tries to explain Euclid's omission to define the circumference. The definition itself contained nothing new in substance. Plato {Parme- nides 137 e) says : " Round is, I take it, that the extremes of which are every way equally distant from the middle " (aTpoyyvkor yi irov Am touto, oi &v to lo^ara irayra^ airo rov pitrnv Ixrov lirri^tj). In Aristotle we find the following expressions : " the circular (v4pi<i>fp6ypajifu>v) plane figure Dounded by one line" {De taeh n. 4, 286 b 13 — 16); "the plane equal (i.e. extending equally all ways) from the middle " (hrUtfov to i*. tov p.iao\s urov), meaning a circle (Rhetoric 111. 6, 1407 b 27); he also contrasts with the circle "any other figure which has not the lines from the middle equal, as for example an egg-shaped figure" {De eaelo n. 4, 287 a 19). The word "centre" {nivrpov) was also regularly used : cf. Produs' quotation from the " oracles " (Wyto), " the centre from which all (lines extending) as far as the rim are equal." The definition as it stands has no genetic character. It says nothing as to the existence or non-existence of the thing defined or as to the method of constructing it. It simply explains what is meant by the word " circle," and is a provisional definition which cannot be used until the existence of circles is proved or assumed. Generally, in such a case, existence is proved by actual construction ; but here the possibility of constructing the circle as defined, and consequently its existence, are postulated (Postulate 3). A genetic definition might state that a circle is the figure described when a straight line, always remaining in one plane, moves about one extremity as a fixed point until it returns to its first position (so Heron, Def. ay). Simplicius indeed, who points out that the distance between the feet of a pair of compasses is a straight line from the centre to the circumference, will have it that Euclid intended by this definition to show how to construct a circle by the revolution of a straight line about one end as centre ; and an- Nairlzi points to this as the explanation (r) of Euclid's definition of a circle as a plane figure, meaning the whole surface bounded by the circumference, and not the circumference itself, and (2) of his omission to mention the " circumference," since with this construction the circumference is not drawn separately as a line. But it is not necessary to suppose that Euclid himself did more than follow the traditional view ; for the same conception of the circle as a plane figure appears, as we have seen, in Aristotle. While, however, i. Deff. 15-17] NOTES ON DEFINITIONS 15—17 185 Euclid is generally careful to say the "circumference of a circle " when he means the circumference, or an arc, only, there are cases where "circle" means "circumference of a circle," e.g. in ill. 10 1 "A circle does not cut a circle in more points than two." Heron, Proclus and Simplicius are all careful to point out that the centre is not the only point which is equidistant from all points of the circumference. The centre is the only point in the plane of the circle ("lying within the figure," as Euclid says) of which this is true; any point not in the same plane which is equidistant from all points of the circumference is a pole. If you set up a "gnomon " (an upright stick) at the centre of a circle (i.e. a line through the centre perpendicular to the plane of the circle), its upper extremity is a pole (Proclus, p. 153, 3); the perpendicular is the locus of all such poles. Definition 17. ktAfitrpof Si rov kvkKuv ttrrlv tvOttOr tls &ia tw Ktvrpov Tjyfj&vrj k<u irtpaiav- p-ivT) ($ tKfXTfpa tq pip7j uiro tt|s ro? kvkXov ircpM^rpfiav, Tfrc? *a>. Stya Ti/ivfi tov kvx\qv. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. The last words, literally " which (straight line) also bisects the circle," are omitted by Simson and the editors who followed him. But they are necessary even though they do not " belong to the definition " but only express a property of the diameter as defined. For, without this explanation, Euclid would not have been justified in describing as a .fcrjw-circle a portion of a circle bounded by a diameter and the circumference cut off by it. Simplicius observes that the diameter is so called because it passes through the whole surface of a circle as if measuring it, and also because it divides the circle into two equal parts. He might however have added that, in general, it is a line passing through a figure where it is widest, as well as dividing it equally : thus in Aristotle ri nara Sia/ttrpw KtCptva, " things diametrically situated " in space, are at their maximum distance apart. Diameter was the regular word in Euclid and elsewhere for the diameter of a square, and also of a parallelogram; diagonal (Stayuptot) was a later term, defined by Heron (Def. by] as the straight line drawn from an angle to an angle. Proclus (p. 157, 10) says that Thales was the first to prove that a circle is bisected by its diameter; but we are not told how he proved it. Proclus gives as the reason of the property "the undeviating course of the straight line through the centre " (a simple appeal to symmetry), but adds that, if it is desired to prove it mathematically, it is only necessary to imagine the diameter drawn and one part of the circle applied to the other ; it is then clear that they must coincide, for, if they did not, and one fell inside or outside the other, the straight lines from the centre to the circumference would not all be equal : which is absurd. Saccheri's proof is worth quoting. It depends on three " Lemmas " immediately preceding, (1) that two straight lines cannot enclose a space, (z) that two straight lines cannot have one and the same segment common, (3) that, if two straight lines meet at a point, they do not touch, but cut one another, at it. " Let MDHNKM be a circle, A its centre, MN a diameter. Suppose i86 BOOK I [i. Deff. 17, tS the portion MNKM of the circle turned about the fixed points M, N, so that it ultimately comes near to or coincides with the remaining portion MNHDM. "Then (i) the whole diameter MAN', with all its points, clearly remains in the 3ame position, since otherwise two straight lines would enclose a space (contrary to the first Lemma). " (ii) Clearly no point K of the circumference NKM falls within or outside the surface enclosed by the diameter MANand the other part, NHDM, of the circumference, since otherwise, contrary to the nature of the circle, a radius as AK would be less or greater than another radius as All. " (iii) Any radius MA can clearly be rectilineally produced only along a single other radius AN, since otherwise (contrary to the second Lemma) two lines assumed straight, e.g. MAN, MAH, would have one and the same common segment. " (iv) All diameters of the circle obviously cut one another in the centre (Lemma 3 preceding), and they bisect one another there, by the general properties of the circle. " From all this it is manifest that the diameter MAN divides its circle and the circumference of it just exactly into two equal parts, and the same may be generally asserted for every diameter whatsoever of the same circle ; which was to be proved." Simson observes that the property is easily deduced from tit. 31 and 24 ; for it follows from lit. 31 that the two parts of the circle are "similar segments" of a circle (segments containing equal angles, in. Def. 11), and from tti. 24 that they are equal to one another. Definition 18. H/iuruKXiov hi itTTt to Trtptt^ofitvov tr)(fffia. inrd re t?/s BiafUTpov Kat Trj% Ivokafi^awo/Ltv^^ iiir avr^e vtpi^tpiia^. nivTfujv Si tov ijfiiKvtt\{oii to avrd, & Ktll TOV KVtikrtv CffTtV. A semicircle if the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle. The last words, "And the centre of the semicircle is the same as that of the circle," are added from Proclus to the definition as it appears in the MSS. Scarburgh remarks that a semicircle has no centre, properly speaking, and thinks that the words are not Euclid's, but only a note by Proclus. I am however inclined to think that they are genuine, if only because of the very futility of an observation added by Proclus. He explains, namely, that the semicircle is the only plane figure that has its centre on its perimeter (!), "so that you may conclude that the centre has three positions, since it may be within the figure, as in the case of a circle, or on the perimeter, as with the semicircle, or outside, as with some conic lines (the single-branch hyperbola presumably)" ! Proclus and Simplicius point out that, in the order adopted by Euclid for these definitions of figures, the first figure taken is that bounded by on* line (the circle), then follows that bounded by two lines (the semicircle), then the triangle, bounded by three lines, and so on. Proclus, as usual, distinguishes I. Deff. 18-ai] NOTES ON DEFINITIONS 17—31 187 different kinds of figures bounded by two lines (pp. 159, 14 — 160, 9). Thus they may be formed (1) by circumference and circumference, e.g. (a) those forming angles, as a tunc (to pijvottSn) and the figure included by two arcs with convexities outward, and (b) the angle-less (iytinov), as the figure included between two concentric circles (the coronal) ; (2) by circumference and straight line, e.g. the semicircle or segments of circles (tty£S<t is a name given to those less than a semicircle); (3) by " mixed " line and " mixed " line, e.g. two ellipses cutting one another ; (4) by " mixed " line and circumference, e.g. intersecting ellipse and circle; (5) by " mixed" line and straight line, e.g. half an ellipse. Following Def. 18 in the mss. is a definition of a segment of a circle which was obviously interpolated from in. Def. 6. Proclus, Martianus Capella and Boethius do not give it in this place, and it is therefore properly omitted. Definitions 19, 20, 21. 19. Sx^fOTo tiSvypa/ifia i<m ra wro tifituSv vepn^optva, rptirXtvpa fiir ra viro rpcwv, TtrpawXtvpa Si Ttt vro Tftrtrapwv, Trokvirktvpa Si to Wd ir\<toi*iii' 9 Ttwdpwv (vfyii'vi- irtpttxoprtva, 20. Tw Si Tparktvpttiv tTxypATwv Uroirktvpov piy tprMMW cart to rat rptts uras tX 01 ' irAtvpas, to-oo-«€ Acs Si to ras Svo povas tcras %\ov irXtvpai, tTKttX.i/fvov Si TO rac rptls dvurow; t\oc vktvpos. 21. *Ett Si Tali' Tpnrktvpwv &)ri]pf£-rwv 6p$oytovtav ptv rpiytovov itrri to l)(ov 6p$ijv yomav, dp.ftkvytii'tQV Si to tyov dfiflkiiav ytortav, o^vycuviO*' Si to Tat Tpits o£cia¥ i%Qv ytin-tas. 19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has Us three sides unequal. x 1 . Further, of trilateral figures, a right-angled triangle it that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute. 1 9. The latter part of this definition, distinguishing three-sided, four-sided and many-sided figures, is probably due to Euclid himself, since the words TplirXtvpov, TtrpairKivpov and ircAvTrAtvpop do not appear in Plato or Aristotle (only in one passage of the Mechanics and of the Problems respectively does even rtrpdirktvpov, quadrilateral, occur). By his use of TtTpa'irAtupoK, quadrilateral, Euclid seems practically to have put an end to any ambiguity in the use by mathematicians of the word Terpdyiiivov, literally "four-angled (figure)," and to have got it restricted to the square. Cf. note on Def. 22, 20. Isosceles (io-octmX^, with equal legs) is used by Plato as well as Aristotle. ScaJene (o-praAijvos, with the variant o-mtA-irtTje) is used by Aristotle of a triangle with no two sides equal : cf. also Tim. Locr. 98 b. Plato, Euthyphro 1 2 rj, i88 BOOK I [i. Deff. 20, 21 applies the term " scalene " to an odd number in contrast to " isosceles " used of an even number, l'roclus (p. 168, 24) seems to connect it with tntatu, to limp ; others make it akin to a-ioAto's, crooked, aslant. Apollonius uses the same word " scalene " of an oblique circular cone. Triangles are classified, first with reference to their sides, and then with reference to their angles. l'roclus points out that seven distinct species of triangles emerge: (1) the equilateral triangle, (2) three species of isosceles triangles, the right-angled, the obtuse-angled and the acute-angled, (3} the same three varieties of scalene triangles. Proclus gives an odd reason for the dual classification according to sides and angles, namely that Euclid was mindful of the fact that it is not every triangle that is trilateral also. He explains this statement by reference (p. 165, 2j) to a figure which some called barb-like (&«SmiSiji) while Zenodovus called it hollow-angled ((toiAoywi'w). Proclus mentions it again in his note on 1. 22 (p. 328, 21 sqq.) as one of the paradoxes of geometry, observing that it is seen in the figure of that proposition. This " triangle " is merely a quadrilateral with a re-entrant angle ; and the idea that it has only three angles is due to the non-recognition of the fourth angle (which is greater than two right angles) as being an angle at all. Since Proclus speaks of the four-sided triangle as " one of the paradoxes in geometry," it is perhaps not safe to assume that the misconception underlying the expression existed in the mind of Proclus alone ; but there does not seem to be any evidence that Zenodorus called the figure in question a triangle (cf. Pappus, ed. Hultsch, pp. 1 154, 1206). Definition 22. Twv Si TrrpawXcvpwv trj(jjfjidTmv Ttrpdymvov p\kv i<mv, S itroTrXtvpoy t4 lart Kai 6p$oy<wtov, irtpofjiTjitts £i, u 6pBoywViov firv, ovx urowktvpov S«, pojiftot S(. S UronXtvpov piv, oIk &p$<rywvutr Si, pop.fio*i£is Bi to t&s i.Trtvarrioir vktvpdx rt xal yinvia^ icac oAAij'Aats ^X ov > * our* i<jQTr\cvpov i<rriv out* ApGoyvviov To 04 Trapa TauTa TfrpdirXtvpa TpaW£ia naKtta&w. 0/ quadrilateral figures, a square is that which is both equilateral and right- angled; an oblong thai which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled ; and a rhomboid that which has its opposite sides and angles equal te one another but is neither equilateral nor right-angled. And let quadrilaterals ether than these be called trapezia. Ttrpayiiivov was already a square with the Pythagoreans (cf, Aristotle, Metapk. 986 a 26), and it is so most commonly in Aristotle ; but in Dt anima 11. 3, 414 t> 31 it seems to be a quadrilateral, and in Metapk. 1054 b 2, " equal and equiangular Ttrpdyaiva," it cannot be anything else but quadri- lateral if "equiangular" is to have any sense. Though, by introducing Tcrpdir\tvpav for any quadrilateral, Euclid enabled ambiguity to be avoided, there seem to be traces of the older vague use of Ttrpdytowr in much later writers. Thus Heron (Def. 100) speaks of a cube as "contained by six equi- lateral and equiangular Ttrpdywra" and Proclus (p. 166, 10) adds to his remark about the " four-sided triangle " that " you might have rtrpaybiva with more than the four sides," where rtrpayvsva can hardly mean squares. tTtpofiyKn, oblong (with sides of different length), is also a Pythagorean term. The word right-angled {ipBoyiiytor) as here applied to quadrilaterals must mean rectangular (i.e., practically, having all its angles right angles) ; for, although it is tempting to take the word in the same sense for a I. Def. u] NOTES ON DEFINITIONS so— a j 189 square as for a triangle (i.e. " having one right angle "), this will not do in the case of the oblong, which, unless it were stated that three of its angles are right angles, would not be sufficiently defined. If it be objected, as it was by Todhunter for example, that the definition of a square assumes more than is necessary, since it is sufficient that, being equilateral, it should have one right angle, the answer is that, as in other cases, the superfluity does not matter from Euclid's point of view ; on the contrary, the more of the essential attributes of a thing that could be included in its definition the better, provided that the existence of the thing defined and its possession of all those attributes is proved before the definition is. actually used ; and Euclid does this in the case of the square by construction in 1. 46, making no use of the definition before that proposition. The word rhombus (po/t/Jot) is apparently derived from fiipfioi, to turn round and round, and meant among other things a spinning-top. Archimedes uses the term solid rhombus to denote a solid figure made up of two right cones with a common circular base and vertices turned in opposite directions. We can of course easily imagine this solid generated by spinning; and, if the cones were equal, the section through the common axis would be a plane rhombus, which would also be the apparent form of the spinning solid to the eye. The difficulty in the way of supposing the plane figure to have been named after the solid figure is that in Archimedes the cones forming the solid are not necessarily equal. It is however possible that the solid to which the name was originally given was made up of two equal cones, that the plane rhombus then received its name from that solid, and that Archimedes, in taking up the old name again, extended its signification (cf. J. H. T. Miiller, Beitrdge zur Terminologie der griechisehen Mathematiker, i860, p. 20). Proclus, while he speaks of a rhombus as being like a shaken, i.e. deformed, square, and of a rhomboid as an oblong that has been moved, tries to explain the rhombus by reference to the appearance of a spinning square {rsrpaymvov poftfioiittvor). It is true that the definition of a rhomboid says more than is necessary in describing it as having its opposite sides and angles equal to one another. The answer to the objection is the same as the answer to the similar objection to the definition of a square. Euclid makes no use in the Elements of the oblong, the rhombus and the rhomboid. The explanation of his inclusion of definitions of these figures is no doubt that they were taken from earlier text- books. From the words "let quadrilaterals other than these be called trapezia" we may perhaps infer that trapezium was a new name or a new application of an old name. As Euclid has not yet defined parallel lines and does not anywhere define a parallelogram, he is not in a position to make the more elaborate classification of quadrilaterals attributed by Proclus to Posidonius and appearing also in Heron's Definitions. It may be shown by the following diagram, distinguishing seven species of quadrilaterals. Quadrilaterals parallelograms non- parallelograms rectangular non-rectangular two sides parallel no sides parallel [traptxtum] {traprt&t) r-U squart oblong rhombus rhomboid i iGscdts trapezium scaitnt trapezium 190 BOOK I [1. Bepf. 2s, a3 It will be observed that, while Euclid in the above definition classes as trapezia all quadrilaterals other than squares, oblongs, r ho in hi, and rhomboids, the word is in this classification restricted to quadrilaterals having two sides (only) parallel, and trapezoid is used to denote the rest Euclid appears to have used trapezium in the restricted sense of a quadrilateral with two sides parallel in his book, vtpi Suuptatav (on divisions of figures). Archimedes uses it in the same sense, but in one place describes it more precisely as a trapezium with its two sides parallel. Definition 23. IlapaX^7]\ai turiv ciStitu, a*riMf iv to) uurw hrtwi&o ov&at mil «*/3aXAo/j.*i'ai tit irnpov itji' imiTtpa. Ta /Mpf in ftrfiirtpa tni/iviirTOtxTiv &XXykiuf. Parallel straight tines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. Wap6XKvj\\K (alongside one another) written in one word does not appear in Plato ; but with Aristotle it was already a familiar term. «(? aire tpok cannot be translated " to infinity " because these words might seem to suggest a region or place infinitely distant, whereas •« airttpoy, which seems to be used indifferently with «jt' Zirnpuv, is adverbial, meaning "without limit," i.e. "indefinitely." Thus the expression is used of a magnitude being "infinitely divisible," or of a series of terms extending without limit in both directions, i<ft' tKattpa ™ fitpift literally "towards both the parts" where "parts" must be used in the sense of "regions" (cf Thuc. 11. 96). It is clear that with Aristotle the general notion of parallels was that of straight lines which do not meet, as in Euclid : thus Aristotle discusses the question whether to think that parallels do meet should be called a geometrical or an ungeometrical error (Anal. post. 1. 12, 77 b 22), and (more interesting still in relation to Euclid) he observes that there is nothing surprising in different hypotheses leading to the same error, as one might conclude that parallels meet by starting from the assumption, either (a) that the interior (angle) is greater than the exterior, or (b) that the angles of a triangle make up more than two right angles (Anal, prior, u. 17, 66 a 11). Another definition is attributed by Prod us to Posidonius, who said that "parallel tines are those which, (being) m one plane, neither converge nor diverge, but have all the perpendiculars equal which are drawn from the points oj one line to tlit other, while such (straight lines) as make the perpendiculars less and less continually do converge to one another ; for the perpendicular is enough to define (opt'f %%v jvrarcu) the heights of areas and the distances between lines. For this reason, when the perpendiculars are equal, the distances between the straight lines are equal, but when they become greater and less, the interval is lessened, and the straight lines converge to one another in the direction in which the less perpendiculars are " (Proclus, p. 176, 6 — 17). Posidonius' definition, with the explanation as to distances between straight lines, their convergence and divergence, amounts to the definition quoted by Simplicius (an-Nairlzi, p. 25, ed. Curt/.e) which described straight lines as parallel if, when they are produced indefinitely both ways, the distance between them, or the perpendicular drawn from either of them to the other, is always equal and not different. To the objection that it should be proved that the distance between two parallel lines is the perpendicular to them Simplicius I. Dtr. j 3 ] NOTES ON DEFINITIONS 21, 13 191 replies that the definition will do equally well if all mention of the perpen- dicular be omitted and it be merely stated that the distance remains equal, although " for proving the matter in question it is necessary to say that one straight line is perpendicular to both" (an-NairizI, ed. Besthorn-Heiberg, p. 9). He then quotes the definition of "the philosopher Aganis": "Parallel straight tines are straight lints, situated in the same plane, the distance between which, if they are predated indefinitely in both directions at the same time, is everywhere the same," (This definition forms the basis of the attempt of "Aganis" to prove the Postulate of Parallels.) On the definition Simplicius remarks that the words "situated in the same plane" are perhaps unnecessary, since, if the distance between the lines is everywhere the same, and one does not incline at all towards the other, they must for that reason be in the same plane. He adds that the "distance" referred to in the definition is the shortest tine which joins things disjoined. Thus, between point and point, the distance is the straight line joining them ; between a point and a straight line or between a point and a plane it is the perpendicular drawn from the point to the tine or plane; "as regards the distance between two lines, that distance is, if the lines are parallel, one and the same, equal to itself at al! places on the lines, it is the shortest distance and, at all places on the lines, perpendicular to both" {ibid, p, 10). The same idea occurs in a quotation by Proclus (p, 177, n) from Geminus. As part of a classification of lines which do not meet he observes : " Of lines which do not meet, some are in one plane with one another, others not. Of those which meet and are in one plane, some are always the same distance from one another, others lessen the distance continually, as the hyper- bola (approaches) the straight line, and the conchoid the straight line (i.e. the asymptote in each case). For these, while the distance is being continually lessened, are continually (in the position of) not meeting, though they converge to one another; they never converge entirely, and this is the most paradoxical theorem in geometry, since it shows that the convergence of some lines is non- convergent. But of lines which are always an equal distance apart, those which are straight and never make the (distance) between them smaller, and which are in one plane, are parallel." Thus the equidistance-theary of parallels (to which we shall return) is very fully represented in antiquity. I seem also to see traces in Greek writers of a conception equivalent to the vicious direction-theory which has been adopted in so many modem text-books. Aristotle has an interesting, though obscure, allusion in Anal, prior, ti. 16, 65 a 4 to a petit io principii committed by "those who think that they draw parallels " (or " establish the theory of parallels," which is a possible translation of t&s «-apaAA»J*ovs ypdifttw) : "for they un- consciously assume such things as it is not possible to demonstrate if parallels do not exist" It is clear from this that there was a vicious circle in the then current theory of parallels ; something which depended for its truth on the properties of parallels was assumed in the actual proof of those properties, e.g. that the three angles of a triangle make up two right angles. This is not the case in Euclid, and the passage makes it clear that it was Euclid himself who got rid of the petitio principii in earlier text-books by formulating and premising before 1. 19 the famous Postulate 5, which must ever be regarded as among the most epoch-making achievements in the domain of geometry. But one of the commentators on Aristotle, Philoponus, has a note on the above passage purporting to give the specific character of the petitio principii alluded to; and it is here that a direction-theory of parallels may be hinted at, i9i BOOK I [i. Def. 12 whether Philoponus is or is not right in supposing that this was what Aristotle had in mind. Philoponus says: "The same thing is done by those who draw parallels, namely begging the original question ; for they will have it that it is possible to draw parallel straight lines from the meridian circle, and they assume a point, so to say, falling on the plane of that circle and thus they draw the straight lines. And what was sought is thereby assumed; for he who does not admit the genesis of the parallels will not admit the point referred to either." What is meant is, I think, somewhat as follows. Given a straight line and a point through which a parallel to it is to be drawn, we are to suppose the given straight line placed in the plane of the meridian. Then we are told to draw through the given point another straight line in the plane of the meridian (strictly speaking it should be drawn in a plane parallel to the plane of the meridian, but the idea is that, compared with the sue of the meridian circle, the distance between the point and the straight line is negligible) ; and this, as I read Philoponus, is supposed to be equivalent to assuming a very distant point in the meridian plane and joining the given point to it. But obviously no ruler would stretch to such a point, and the objector would say that we cannot really direct a straight line to the assumed distant point except by drawing it, without more ado, parallel to the given straight line. And herein is the pditio principii. I am confirmed in seeing in Philoponus an allusion to a direction-theory by a remark of Schotten on a similar reference to the meridian plane supposed to be used by advocates of that theory. Schotten is arguing that direction is not in itself a conception such that you can predicate one direction of two different lines. " If any one should reply that nevertheless many lines can be conceived which all have the direction from north to south," he replies that this represents only a nominal, not a real, identity of direction. Coming now to modern times, we may classify under three groups practically all the different definitions that have been given of parallels (Schotten, op. cit. it p. 188 sqq.). (i) Parallel straight lines have no point common, under which general conception the following varieties of statement may be included : (a) they do not cut one another, (6) they meet at infinity, or (c ) they have a common point at infinity. (a) Parallel straight lines have the same, or like, direction or directions, under which class of definitions must be included all those which introduce transversals and say that the parallels make equal angles with a transversal. (3) Parallel straight lines have the distance between them constant; with which group we may connect the attempt to explain a parallel as the geometrical focus of all points which are equidistant from a straight line. But the three points of view have a good deal in common ; some of them lead easily to the others. Thus the idea of the lines having no point common led to the notion of their having a common point at infinity, through the influence of modem geometry seeking to embrace different cases under one conception ; and then again the idea of the lines having a common point at infinity might suggest their having the same direction. The " non-secant " idea would also naturally lead to that of equidistance (3}, since our observation shows that it is things which come nearer to one another that tend to meet, and hence, if lines are not to meet, the obvious thing is to see that they shall not conic nearer, i.e. shall remain the same distance apart. L Def. i 3 ] NOTE ON DEFINITION 13 193 We will now take the three groups in order. (1) The first observation of Schotten is that the varieties of this group which regard parallels as (a) meeting at infinity or (6) having a common point at infinity (first mentioned apparendy by Kepler, 1604, as a " facpn de parler " and then used by Desargues, 1630) are at least unsuitable definitions for elementary text-books. How do we know that the lines cut or meet at infinity ? We are not entitled to assume either that they do or that they do not, because "infinity" is outside our field of observation and we cannot verify either. As Gauss says (letter to Schumacher), " Finite man cannot claim to be able to regard the infinite as something to be grasped by means of ordinary methods of observation." Steiner, in speaking of the rays passing through a point and successive paints of a straight tine, observes that as the point of intersection gets further away the ray moves continually in one and the same direction (" nach einer und derselben Richtung hin ") ; only in one position, that in which it is parallel to the straight line, "there is no real cutting" between the ray and the straight line ; what we have to say is that the ray is "directed towards the infinitely distant point on the straight line." It is true that higher geometry has to assume that the lines do meet at infinity: whether such lines exist in nature or not does not matter (just as we deal with "straight lines " although there is no such thing as a straight line). But if two lines do not cut at any finite distance, may not the same thing be true at infinity also? Are lines conceivable which would not cut even at infinity but always remain at the same distance from one another even there? Take the case of a line of railway. Must the two rails meet at infinity so that a train could not stand on them there (whether we could see it or not makes no difference)? It seems best therefore to leave to higher geometry the conception of infinitely distant points on a line and of two straight lines meeting at infinity, like imaginary points of intersection, and, for the purposes of elementary geometry, to rely on the plain distinction between " parallel " and " cutting " which average human intelligence can readily grasp. This is the method adopted by Euclid in his definition, which of course belongs to the group (1) of definitions regarding parallels as non-secant. It is significant, I think, that such authorities as Ingram! {Elementi di geometria, 1904) and Enriques and Amaldi {EUmenti di geometria, 1905), after all the discussion of principles that has taken place of late years, give definitions of parallels equivalent to Euclid's : " those straight lines in a plane which have not any point in common are called parallels." Hilbert adopts the same point of view, Veronese, it is true, takes a different line. In his great work Eondamenti di geometria, 1891, he had taken a ray to be parallel to another when a point at infinity on the second is situated on the first ; but he appears to have come to the conclusion that this definition was unsuitable for his Element i. He avoids however giving the Euclidean definition of parallels as "straight lines in a plane which, though produced indefinitely, never meet," because no one has ever seen two straight lines of this sort," and because the postulate generally used in connexion with this definition is not evident in the way that, in the field of our experience, it is evident that only one straight line can pass through two points. Hence he gives a different definition, for which he claims the advantage that it is independent of the plane. It is based on a definition of figures "opposite to one another with respect to a point" (or rtflex figures). "Two figures are opposite to one another with respect to a point O, e.g. the figures ABC ... and A'ffC .,., if to every point of the one there corresponds one sole point of the other, and if the segments 194 BOOK I [1. Dbf. 13 OA, OB, OC, ... joining the points of one figure to O are respectively equal and opposite to the segments OA\ OB, OC",... joining to O the corresponding points of the second " : then, a transversal of two straight lines being any segment having as its extremities one point of one line and one point of the other, " two straight lines are called parallel if one of them contains two points opposite to two paints of the other with respect to the middle point of a common transversal" It is true, as Veronese says, that the parallels so defined and the parallels of Euclid are in substance the same ; but it can hardly be said that the definition gives as good an idea of the essential nature of parallels as does Euclid's. Veronese has to prove, of course, that his parallels have no point in common, and his "Postulate of Parallels" can hardly be called more evident than Euclid's : " If two straight lines are parallel, they are figures opposite to one another with respect to the middle points of all their transversal segments." (2) The direction-theory. The fallacy of this theory has nowhere been more completely exposed than by C. L. Dodgson (Euclid and his modern Rivals, 1879). According to Killing (JEinfiihrung in die Grundlagen der Geometrie, 1. p. 5) it would appear to have originated with no less a person than Leibniz. In the text-books which employ this method the notion of direction appears to be regarded as a primary, not a derivative notion, since no definition is given. But we ought at least to know how the same direction or like directions can be recognised when two different straight lines are in question. But no answer to this question is forthcoming. The fact is that the whole idea as applied to non- coincident straight lines is derived from knowledge of the properties of parallels ; it is a case of explaining a thing by itself. The idea of parallels being in the same direction perhaps arose from the conception of an angle as a difference of direction (the hoi low n ess of which has already been exposed) ; sameness of direction for parallels follows from the same " difference of direction " which both exhibit relatively to a third line. But this is not enough. As Gauss said ( Werke, iv. p. 365), " If it [identity of direction] is recognised by the equality of the angles formed with one third straight line, we do not yet know without an antecedent proof whether this same equality will also be found in the angles formed with a fourth straight line " {and any number of other transversals) ; and in order to make this theory of parallels valid, so far from getting rid of axioms such as Euclid's, you would have to assume as an axiom what is much less axiomatic, namely that " straight lines which make equal corresponding angles with a certain transversal do so with any transversal " (Dodgson, p. 101). (3) In modern times the conception of parallels as equidistant straight lines was practically adopted by Clavius (the editor of Euclid, born at Bamberg, 1537) and (according to Saccheri) by Borelli (Euctides restitutus, 1658) although they do not seem to have defined parallels in this way. Saccheri points out that, before such a definition can be used, it has to be proved that " the geometrical locus of points equidistant from a straight line is a straight line." To do him justice, Clavius saw this and tried to prove it : he makes out that the locus is a straight line according to the definition of Euclid, because " it lies evenly with respect to all the points on it"; but there is a confusion here, because such "evenness" as the locus has is with respect to the straight line from which its points are equidistant, and there is nothing to show that it possesses this property with respect to itself. In fact the theorem cannot be proved without a postulate. I. Post, i] NOTE ON POSTULATE t 195 Postulate i. 'Hmjff&n aire jravres aijfuiou irl woe (njpMOV tvSiiav ypaptpTJv dyaytiv. Let the following be postulated : to draw a straight line from any point to any point. From any point to any point. In general statements of this kind the Greeks did not say, as we do, " any point," " any triangle " etc., but "every point," " every triangle " and the like. Thus the words are here literally "from every point to every point." Similarly the first words of Postulate 3 are " with every centre and distance," and the enunciation, e.g., of 1. 18 is " In every triangle the greater side subtends the greater angle." It will be remembered that, according to Aristotle, the geometer must in general assume what a thing is, or its definition, but must prove that it is, i.e. the existence of the thing corresponding to the definition : only in the case of the two most primary things, points and lines, does he assume, without proof, both the definition and the existence of the thing defined. Euclid has indeed no separate assumption affirming the existence oi points such as we find nowadays in text-books like those of Veronese, Ingrami, Enriques, "there exist distinct points" or "there exist an infinite number of points." But, as re- gards the only lines dealt with in the Elements, straight lines and circles, existence is asserted in Postulates 1 and 3 respectively. Postulate 1 however does much more than (1) postulate the existence of straight lines. It is (2) an answer to a possible objector who should say that you cannot, with the imperfect instruments at your disposal, draw a mathematical straight line at all, and consequently (in the words of Aristotle, A/ml. post. 1. to, 76 b 41) that the geometer uses false hypotheses, since he calls a line a foot long when it is not or straight when it is not straight. It would seem (if Gherard's translation is right) that an-Nairlsd saw that one purpose of the Postulate was to refute this criticism : "the utility of the first three postulates is (to ensure) that the weakness of our equipment shall not prevent (scientific) demonstration " (ed. CurUe, p. 30). The fact is, as Aristotle says, that the geometer's demon- stration is not concerned with the particular imperfect straight line which he has drawn, but with the ideal straight line of which it is the imperfect representation. Simplidus too indicates that the object of the Postulate is rather to enable the drawing of a mathematical straight line to be imagined than to assert that it can actually be realised in practice : " he would be a rash person who, taking things as they actually are, should postulate the drawing of a straight line from Aries to Libra." There is still something more that must be inferred from the Postulate combined with the definition of a straight line, namely {3) that the straight line joining two points is unique : in other words that, if two straight lines ("rectilineal segments," as Veronese would call them) have the same extremities, they must coincide throughout their length. The omission of Euclid to state this in so many words, though he assumes it in 1. 4, is no doubt answerable for the interpolation in the text of the equivalent assumption that two straight lines cannot enclose a space, which has constantly appeared in mss. and editions of Euclid, either among Axioms or Postulates. That Postulate 1 included it, by conscious implication, is even clear from Proclus' words in his note on 1. 4 (p. 139, 16) : "therefore two straight lines do not enclose a space, and it was with knowledge of this fact that the writer of the Elements said in the first of his Postulates, to draw a straight line from any point to any point, implying that it is one straight line which would always join the two points, not two." t9 6 BOOK I [i. Post, i, * Proclus attempts in the same note (p. 339) to prove that two straight lines cannot enclose a space, using as his basis the definition of the diameter of a circle and the theorem, stated in it, that any diameter divides the circle into two equal parts. Suppose, he says, ACS, ADB to be two straight lines enclosing a space. Produce them (beyond B) indefinitely. With centre £ and distance AB describe a circle, cutting the lines so produced in F, E respectively. Then, since ACBF, ADBE are both diameters cutting off semi-circles, the arcs AE, AEF are equal : which is impossible. Therefore etc It will be observed, however, that the straight lines produced are assumed to meet the circle given in two different points E, F, whereas, for anything we know, E, F might coincide and the straight lines have three common points. The proof is therefore delusive. Saccheri gives a different proof. From Euclid's definition of a straight line as that which lies evenly with its points he infers that, when such a line is turned about its two extremities, which remain fixed, all the points on it must remain throughout in the same position, and cannot take up different positions as the revolution proceeds. "In this view of the straight line the truth of the assertion that two straight lines do not enclose a space is obviously involved. In fact, if two lines are given which enclose a space, and of which the two points A and X are the common extremities, it is easily shown that neither, or else only one, of the two lines is straight." It is however better to assume as a postulate the (act, inseparably connected with the idea of a straight line, that there exists only one straight line containing two given points, or, if two straight lines have two points in common, they coincide throughout. Postulate 2. Kai vertpao-iiinpr ivOtlav Kara to minuet <V tiStiai infiaXtuf. To produce a finite straight line continuously in a straight line. I translate ir«r tpaapi vyv by finite, because that is the received equivalent, and because any alternative word such as limited, terminated, if applied to a straight line, would equally fail to express what modem Italian geometers aptly call a rectilineal segment, that is, a straight line having two extremities. Just as Post. 1 asserting the possibility of drawing a straight line from any one point to another must be held to declare at the same time that the straight line so drawn is unique, so Post. 2 maintaining the possibility of producing a finite straight line (a " rectilineal segment ") continuously in a straight line must also be held to assert that the straight line can only be produced in one way at either end, or that the produced part in either direction is unique; in other words, that two straight lines cannot have a common segment, This latter assumption is not expressly appealed to by Euclid until XL t. But it is needed at the very beginning of Book 1. Proclus (p. 114, 18) says that Zeno of Sidon, an Epicurean, maintained that the very Erst proposition 1. 1 requires it to be admitted that " two straight lines cannot have the same segments " ; otherwise AC, BC might meet before they arrive at C and have the rest of their length common, in which case the actual triangle formed by them and A B would not be equilateral. The assumption that two straight lines cannot have a common segment is certainly necessary in 1. 4, where one side of one triangle is placed on that side of the other i. Post, a] NOTES ON POSTULATES i, 2 197 triangle which is equal to it, and it is inferred that the two coincide throughout their length : this would by no means follow if two straight lines could have a common segment. Proclus (p. 315, 24), while observing that Post. 2 clearly indicates that the produced portion must be one, attempts to prove it, but unsuccessfully. Both he and Simplicius practically use the same argument. Suppose, says Proclus, that the straight lines AC, AD have .« as 1 common segment With centre B and radius BA describe a circle (Post. 3) meeting AC, AD in C, D.- Then, since ABC is a straight line through the centre, AEC is a semi-circle. Similarly, ABD being a straight line through the centre, A ED is a semi-circle. Therefore AEC is equal to AED : which is impossible. Proclus observes that Zeno would object to this proof as really depending on the assumption that "two circumferences (of circles) cannot have one portion common " ; for this, he would say, is assumed in the common proof by superposition of the fact that a circle is bisected by a diameter, since that proof takes it for granted that, if one part of the circumference cut off by the diameter, when applied to the other, does not coincide with it, it must neces- sarily fall either entirely outside or entirely inside it, whereas there is nothing to prevent their coinciding, not altogether, but in part only ; and, until you really prove the bisection of a circle by its diameter, the above proof is not valid. Posidonius is represented as having derided Zeno for not seeing that the proof of the bisection of a circle by its diameter goes on just as well if the circumferences fail to coincide in fart only. But the true objection to the proof above given is that the proof of the bisection of a circle by any diameter itself assumes that two straight lines cannot have a common segment; for, if we wish to draw the diameter of a circle which has its extremity at a given point of the circumference we have to join the latter point to the centre (Post. 1) and then to produce the straight line so drawn till it meets the circle again (Post, a), and it is necessary for the proof that the produced part shall be unique. Saccheri adopted the proper order when he gave, first the proposition that two straight lines cannot have a common segment, and after that the proposition that any diameter of a circle bisects the circle and its circumference. Saccheri's proof of the former is very interesting as showing the thorough- ness of his method, if not at the end entirely convincing. It is in five stages which I shall indicate shortly, giving the full argument of the first only. Suppose, if possible, that AX is a common segment of both the straight lines AXB, AXC, in one plane, produced beyond X. Then describe about X as centre, with radius XB or XC, the arc BMC, and draw through X to any point on it the straight line XM. (i) I maintain that, with the assumption made, the lint AXM is also a straight line which is drawn from the point A to the point X and pro- duced beyond X. For, if this line were not straight, we could draw another straight line AM which for its part would be straight. This straight line will either (a) cut one of the two straight lines XB, XC in a certain point A" or (6) enclose one of them, for instance XB, in the area bounded by AX, XM and APLM. i 9 8 BOOK I [i. Post. 2 But the first alternative (a) obviously contradicts the foregoing lemma [that two straight lines cannot enclose a space], since in that case the two lines AXK, ATK, which by hypothesis are straight, would enclose a space. The second possibility (b) is at once seen to involve a similar absurdity. For the straight Jine X B must, when produced beyond 3, ultimately meet APLM in a point L. Consequently the two lines AXBL, APL, which by hypothesis are straight, would again enclose a space. If however we were to assume that the straight line XB produced beyond B will ultimately meet either the straight line XM or the straight line XA in another point, we should in the same way arrive at a contradiction. From this it obviously follows that, on the assumption made, the line AXM is itself the straight line which was drawn from the point A to the point M\ and that is what was maintained. The remaining stages are in substance these. (ii) If the straight line AXB, regarded as rigid, revolves about AX as axis, it cannot assume two more positions in the same plane, so that, for example, in one position XB should coincide with XC, and in the other -with XM, [This is proved by considerations of symmetry. AXB cannot be altogether " similar or equal to " AXC, if viewed from the same side {left or right) of both : otherwise they would coincide, which by hypothesis they do not. But there is nothing to prevent AXB viewed from one side (say the left) being " similar or equal to " AXC viewed from the other side (i.e. the right), so that AXB can, without any change, be brought into the position AXC. AXB cannot however take the position of the other straight line AXM&& well- If they were like on one side, they would coincide ; if they were like on opposite sides, AXM, AXC would be like on the same side and therefore coincide.] (iii) The other positions of AXB during the revolution must be above or below the original plane. (iv) It is next maintained that there is a point D on the are BC such that, if XD is drawn, AXD is not only a straight line but is such that viewed from the left side it is exactly "similar or equal" to what it is when vieioedfrom the right side. [First, it is proved that points M, F can be found on the arc, corresponding in the same way as B, C do, but nearer together, and of course AXM, AXF are both straight lines. Secondly, similar corresponding points can be found still nearer together, and so on continually, until either (a) we come to one point D such that AXD is exactly like itself when the right and left sides are compared, or (b) there are two ultimate points of this sort M, F, so that both AXM, AXF have this property. Thirdly, {&) is ruled out by reference to the definition of a straight line. Hence (a) only is true, and there is only one point D such as described.] (v) Lastly, Saccheri concludes that the straight line AXD so determined " is alone a straight line, and the immediate prolongation from A beyond X to D," relying again on the definition of a straight line as " lying evenly.'' Simson deduced the proposition that two straight lines cannot have a common segment as a corollary from 1, 11; but his argument is a complete petitio principii, as shown by Tod hunter in his note on that proposition. Proclus (p. 317, 10) records an ancient proof also based on the proposition 1. 11. Zeno, he says, propounded this proof and then criticised it. t. Post, i, 3] NOTES ON POSTULATES 2, 3 i0o Suppose that two straight tines AC, AD have a common segment AB, and let BE be drawn at right angles to AC. Then the angle EBC is right. If then the angle EBD is also right, the two angles will be equal ; which is impossible. If the angle EBD is not right, draw BE at right angles to AD ; therefore the angle FBA is right. But the angle EBA is right. Therefore the angles EBA, FBA are equal : which is impossible. Zeno objected to this, says Proclus, because it assumed the later pro- position 1. 1 1 for its proof. Posidonius said that there was no trace of such a proof to be found in the text-books of Elements, and that it was only invented by Zeno for the purpose of slandering contemporary geometers. Posidonius maintains further that even this proof has something to be said for it. There must be some straight line at right angles to each of the two straight lines A C, AD (the very definition of right angles assumes this) : "suppose then it happens to be the straight line we have set up." Here then we have an ancient instance of a defence of hypothetical construction, but in such apologetic terms (" it is possible to say something even for this proof") that we may conclude that in general it would not have been accepted by geometers of that time as a legitimate means of proving a proposition. Todhunter proposed to deduce that ftvo straight lines cannot have a common segment from i. 13. But this will not serve either, since, as before mentioned, the assumption is really required for i. 4. It is best to make it a postulate. Postulate 3. Kol itavTi Wirpcii ttal 8ta<mj/iaTL kvuXov ypafaa&at. To describe a circle with any centre and distance. In this case Euclid's text has the passive of the verb : "a circle can be drawn " ; Proclus however has the active {yputyu) as Euclid has in the first two Postulates. Distance, Swo-nJfiaTt. This word, meaning '' distance " quite generally (cf. Arist. Metaph. 1055 a 9 "it is between extremities that distance is greatest," ibid. 1056 a 36 " things which have something between them, that is, a certain distance "), and also " distance " in the sense of " dimension " (as in " space has three dimensions, length, breadth and depth," Arist. Physics iv. 1, 209 a 4), was the regular word used for describing a circle with a certain radius, the idea being that each point of the circumference was at that distance from the centre (cf. Arist Afetearologica ill. 5, 376 b 8 : "if a circle be drawn... with distance Mil "). The Greeks had no word corresponding to radius : if they had to express it, they said "(straight lines) drawn from the centre" («u « tou nfrrpov, Eucl. lit. Def. 1 and Prop. 26 \ Mtteorologica 11.5,362 b! has the full phrase oi in toS kot^ob ayo/MW" ypauftai). Mr Frankland observes that it would be remarkable if, unlike Postulates 1 and 2, this Postulate implied merely what tt says, that a circle can be drawn with any centre and distance. We may regard it, if we please, as helping to the complete delineation of the Space which Euclid's geometry is to investigate formally. The Postulate has the effect of removing any restriction upon the siie of the circle. It may (1) be indefinitely small, and this implies that space is continuous, not discrete, with an irreducible minimum distance between *oo BOOK I [i. Post 3. 4 contiguous points in it (a) The circle may be indefinitely large, which implies the fundamental hypothesis of infinitude of space. This last assumed characteristic of space is essential to the proof of 1. 16, a theorem not universally valid in a space which is unbounded in extent but finite in size. It would however be unsafe to suppose that Euclid foresaw the use to which his Postulate might thus be put, or formulated it with such an intention. Postulate 4. That all fight angles are equal to one another. While this Postulate asserts the essential truth that a right angle is a determinate magnitude so that it really serves as an invariable standard by which other (acute and obtuse) angles may be measured, much more than this is implied, as will easily be seen from the following consideration. If the statement is to be proved, it can only be proved by the method of applying one pair of right angles to another and so arguing their equality. But this method would not be valid unless on the assumption of the invariability of figures, which would therefore have to be asserted as an antecedent postulate. Euclid preferred to assert as a postulate, directly, the fact that all right angles are equal ; and hence his postulate must be taken as equivalent to the principle of invariability of figures or its equivalent, the homogeneity of space. According to Proclus, Geminus held that this Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property of right angles. Proclus further observes (p. i88, 8} that it is not a postulate in Aristotle's sense either. (In this I think he is wrong, as explained above.) Proclus himself, while regarding the assumption as axiomatic ("the equality of right angles suggests itself even hy virtue of our common notions"), is prepared with a proof, if such is asked for. Let ABC, DEF be two right angles. If they are not equal, one of them must be the greater, say ABC. Then, if we apply DE to AB, EF will fall within ABC, as BG. Produce CB to H. Then, since ABC is a right angle, so is ABU, and the two angles are equal (a right angle being by definition equal to its adjacent angle). Therefore the angle ABH is greater than the angle ABG. Producing GB to K, we have similarly the two angles ABK, ABG both right and equal to one another; whence the angle ABH is less than the angle ABG. But it is also greater : which is impossible. Therefore etc. A defect in this proof is the assumption that CB t GB can each be produced only in one way, and that BK falls outside the angle ABH. Saccheri's proof is more careful in that he premises a third lemma in addition to those asserting <i) that two straight lines cannot enclose a space and (*) that two straight lines cannot have a common segment. The third lemma is : If two straight lines AB, CXD meet one another at an intermediate point X, they do not touch at that point, but cut one another. i. Post. 4] NOTES ON POSTULATES 3, 4 *oi M Suppose now that DA standing on BAC makes the two angles DAB, DAC equal, so that each is a right angle by the definition. Similarly, let LHioTm. with the straight line FHM the right angles LHF, LHM. Let DA, HL be equal ; and sup- pose the whole of the second figure so laid upon the first that the point H falls on A, and L on D. Then the straight line FHM 'will (by the third lemma) not touch the straight line BC at A ; it will either (a) coincide exactly with BC, or (0) cut it so that one of its extremities, as F, will fall above [BC] and the other, M, below it. If the alternative (a) is true, we have already proved the exact equality of all rectilineal right angles. Under alternative (e) we prove that the angle I.HF, being equal to the angle DAF, is less than the angle DAB or DAC, and a fortiori less than the angle DAM ox LHM; which is contrary to the hypothesis. [Hence (a) is the only possible alternative, so that all right angles are equal.] Saicheri adds that it makes no difference if the. angle DAF diverges infinitely lift It from the angle DAB. This would equally lead to a conclusion contradicting the hypothesis. It will be observed that Saccheri speaks of "the exact equality of all rectilineal right angles." He may have had in mind the remark of Pappus, quoted by Proclus (p. t8a, 1 1), that the converse of this postulate, namely that an angle which is equal to a right angle is also right, is not necessarily true, unless the former angle is rectilineal. Suppose two equal straight lines BA, BCsX right angles to one another, and semi-circles described on BA, BC respectively as AEB, BDC in the figure. Then, since the semi-circles are equal, they coincide .if applied to one another. Hence the "angles" BBA, DBC are equal. Add to each the " angle " ABD ; and it follows that the iunular angle EBD is equal to the right angle ABC. (Similarly, if BA, BCbt inclined at an acute or obtuse angle, instead of at a right angle, we find a Iunular angle equal to an acute or obtuse angle.) This is one of the curiosities which Greek commentators delighted in. Veronese, Ingrami, and Enriques and Amaldi deduce the fact that all right angles are equal from the equivalent fact that all flat angles are equal, which is either itself assumed as a postulate or immediately deduced from some other postulate. Hilbert takes quite a different line. He considers that Euclid did wrong in placing Post 4 among "axioms." He himself, after his Group in. of Axioms containing six relating to congruence, proves several theorems about the congruence of triangles and angles, and then deduces our Postulate. As to the raison fHre and the place of Post. 4 one thing is quite certain. It was essential from Euclid's point of view that it should come before Post. 5, since the condition in the latter that a certain pair of angles are together less than two right angles would be useless unless it were first made clear that right angles are angles of determinate and invariable magnitude. jo* BOOK I [i. Post. 5 Postulate 5. K HI cav cfc Sue tv@twS t\$<ltt fynriVrovtra T4V ^VTOf Kcii tVt TO. ctvrii ft* 1*7) ytavia s Silo upCuS* Aa<r<rovas iroip, (u^aAAo/itfas To! Sw> tWtiM i»' aiMipw <ri>/ijrtnTMif, c£ a ,r< i,ij^ (tcrtv ai toJv Buo op&uv eX<£crf/in'€!. TXrt/, if a straight line falling on two straight lints make the interior angles on the same side less than two right angles, the two straight lines, if produeed indefinitely, meet on that side on whieh are the angles less than the tit<o right angles. Although Aristotle gives a clear idea of what he understood by a postulate, he does not give any instances from geometry; still less has he any allusion recalling the particular postulates found in Euclid. We naturally infer that the formulation of these postulates was Euclid's own work. There is a more positive indication of the originality of Postulate 5, since in the passage {A rial prior. 11. 16, 65 a 4) quoted above in the note on the definition of parallels he alludes to some petitio principii involved in the theory of parallels 'current in his time. This reproach was removed by Euclid when he laid down this epoch-making Postulate. When we consider the countless successive attempts made through more than twenty centuries to prove the Postulate, many of them by geometers of ability, we cannot but admire the genius of the man who concluded that such a hypothesis, which he found necessary to the validity of his whole system of geometry, was really indemonstrable. From the very beginning, as we know from Proclus, the Postulate was attacked as such, and attempts were made to prove it as a theorem or to get rid of it by adopting some other definition of parallels; while in modem times the literature of the subject is enormous. Riccardi (Saggio di una bibliografia Euclidea, Part iv,, Bologna, 1890) has twenty quarto pages of titles of mono- graphs relating to Post. 5 between the dates 1607 and 1887. Max Simon {Ueber die Entwickit/ng der Elementar-geometrie im XIX. Jakrhundert, 1906) notes that he has seen three new attempts, as late as 1891 (a century after Gauss laid the foundation of non-Euclidean geometry), to prove the theory of parallels independently of the Postulate. Max Simon himself (pp. 53 — 61) gives a large number of references to books or articles on the subject and refers to the copious information, as to contents as well as names, con- tained in Schotten's Inhalt mid Methode des planimetrisehen Unterrichts, 11. PP '83— 33 2 - This note will include some account of or allusion to a few of the most noteworthy attempts to prove the Postulate. Only those of ancient times, as being less generally accessible, will be described at any length ; shorter references must suffice in the case of the modern geometers who have made the most important contributions to the discussion of the Postulate and have thereby, in particular, contributed most towards the foundation of the non- Euclidean geometries, and here I shall make use principally of the valuable Article 8, Sulla teoria delle paraliele e suite geometric non-euclidee (by Roberto Bonola), in Quesiioni riguardanti le matematiehe elementari, I. pp. 147 — 363. Proclus (p. 191, xi sqq.) states very clearly the nature of the first objec- tions taken to the Postulate. " This ought even to be struck out of the Postulates altogether ; for it is a theorem involving many difficulties, which Ptolemy, in a certain book, set himself to solve, and it requires for the demonstration of it a number of definitions as well as theorems. And the converse of it is actually proved by Euclid himself as a theorem. It may be that some would be t. Post. 5] NOTE ON POSTULATE 5 203 deceived and would think it proper to place even the assumption in question among the postulates as affording, in the lessening of the two right angles, ground for an instantaneous belief that the straight lines converge and meet. To such as these Geminus correctly replied that we have learned from the very pioneers of this science not to have any regard to mere plausible imagin- ings when it is a question of the reasonings to be included in our geometrical doctrine. For Aristotle says that it is as justifiable Co ask scientific proofs of a rhetorician as to accept mere plausibilities from a geometer; and Sim mi as is made by Plato to say that he recognises as quacks those who fashion for themselves proofs from probabilities. So in this case the fact that, when the right angles are lessened, the straight lines converge is true and necessary; but the statement that, since they converge more and more as they are pro- duced, they will sometime meet is plausible but not necessary, in the absence of some argument showing that this is true in the case of straight tines. For the fact that some lines exist which approach indefinitely, but yet remain non-secant (itrvpirrtirroi), although it seems improbable and paradoxical, is nevertheless true and fully ascertained with regard to other species of lines. May not then the same thing be possible in the case of straight lines which happens in the case of the lines referred to ? Indeed, until the statement in the Postulate is clinched by proof, the facts shown in the case of other lines may direct our imagination the opposite way. And, though the controversial arguments against the meeting of the straight lines should contain much that is surprising, is there not all the more reason why we should expel from our body of doctrine this merely plausible and unreasoned (hypothesis) ? "It is then clear from this that we must seek a proof of the present theorem, and that it is alien to the special character of postulates. But how it should be proved, and by what sort of arguments the objections taken to it should be removed, we must explain at the point where the writer of the Elements is actually about to recall it and use it as obvious. It will be necessary at that stage to show that its obvious character does not appear independently of proof, but is turned by proof into matter of knowledge." Before passing to the attempts of Ptolemy and Proclus to prove the Postulate, I should note here that Simplicius says (in an-Nairlzi, ed. Besthorn- Heiberg, p. 119, ed. Curtze, p. 65) that this Postulate is by no means manifest, but requires proof, and accordingly " Abthiniathus " and Diodorus had already proved it by means of many different propositions, while Ptolemy also had explained and proved it, using for the purpose Eucl. t. 13, 15 and 16 (or 18). The Diodorus here mentioned may be the author of the Analtmma on which Pappus wrote a commentary. It is difficult even to frame a conjecture as to who "Abthiniathus" is. In one place in the Arabic text the name appears to be written " Anthisathus " (H. Suter in Zeztstkrifi fiir Math, und Phystk, xxxviii,, hist. litt. Abth. p. 194). It has occurred to me whether he might be Peithon, a friend of Serenus of Antinoeia (Antinoupolis) who was long known as Serenus of Antissa. Serenus says (De sittione cyiindri, ed. Heiberg, p. 96): "Peithon the geometer, explaining parallels in a work of his, was not satisfied with what Euclid said, but showed their nature more cleverly by an example; for he says that parallel straight lines are such a thing as we see on walls or on the ground in the shadows of pillars which are made when either a torch or a lamp is burning behind them. And, although this has only been matter of merriment to every one, I at least must not deride it, for the respect I have for the author, who is my friend." If Peithon was known as " of Antinoeia " or " of Antissa," the two forms of the mysterious name might perhaps be an attempt at an equivalent ; but this is no more than a guess. so 4 BOOK I [i. Post. 5 Simplicity adds in full and word for word the attempt of his " friend " or his " master Aganis " to prove the Postulate. Proclus returns to the subject (p, 365, 5) in his note on Eucl. 1. 29. He says that before his time a certain number of geometers had classed as a theorem this Euclidean postulate and thought it matter for proof, and he then proceeds to give an account of Ptolemy's argument. Noteworthy attempts to prove the Postulate. Ptolemy. We learn from Proclus (p. 365, 7 — -11) that Ptolemy wrote a book on the proposition that " straight lines drawn from angles less than two right angles meet if produced," and that he used in his "proof" many of the theorems in Euclid preceding 1. 19. Proclus excuses himself from reproducing the early part of Ptolemy's argument, only mentioning as one of the propositions proved in it the theorem of Eucl. 1. 28 that, if two straight lines meeting a transversal make the two interior angles on the same side equal to two right angles, the straight lines do not meet, however far produced. I. From Proclus' note on I. 28 (p. 362, 14 sq.) we know that Ptolemy proved this somewhat as follows. Suppose that there are two straight lines A B, • CD, and that EFGff, meeting them, makes the angles BFG, FGD equal to two right angles. I say that AB, CD are parallel, that is, they are non-secant. For, if possible, let FB, GD meet at K.' Now, since the angles BFG, FGD are equal to two right angles, while the four angles AFG, BFG, FGD, FGC are together equal to four right angles, the angles AFG, FGC are equal to two right angles. "If therefore FB, GD, when the interior angles are equal to two right angles, meet at K, the straight lines FA, GC will also meet if produced ',- for the angles AFG, CGFare also equal to two right angles. "Therefore the straight lines will either meet in both directions or in neither direction, if the two pairs of interior angles are both equal to two right angles. " Let, then, FA, GC meet at L. "Therefore the straight lines LABK, LCDK enclose a space; which is impossible. " Therefore it is not possible for two straight lines to meet when the interior angles are equal to two right angles. Therefore they are parallel." [The argument in the words italicised would be clearer if it" had been shown that the two interior angles on one side of EH are severally equal to the two interior angles on the other, namely BFG to CGF and FGD to AFG ; whence, assuming FB, GD to meet in K, we can take the triangle KFG and place it (e.g. by rotating it in the plane about O the middle point of FG) so that FG falls where GFis in the figure and GD falls on FA, in which case FB must also fall on GC; hence, since FB, GD meet at K, GC and FA must meet at a corresponding point L, Or, as Mr Frank land does, we may substitute for FG a straight line MN through the middle point of FG drawn perpendicular to one of the parallels, say AB. Then, since the two triangles OMF, ONG have two angles equal respectively, namely FOM to F 7 i. Post. 5] NOTE ON POSTULATE 5 105 GON(i. 15) and OFM to 0GJV; and one side OF equal to one side OG, the triangles are congruent, the angle ONG is a right angle, and MN is perpen- dicular to both AB and CD. Then, by the same method of application, MA, NC are shown to form with MN a triangle MALCN congruent with the triangle NDKBM, and MA, NC meet at a point L corresponding to K. Thus the two straight lines would meet at the two points K, L, This is what happens under the Riemann hypothesis, where the axiom that two straight lines cannot enclose a space does not hold, but all straight lines meeting in one point have another point common also, and e,g. in the particular figure just used K, L are points common to all perpendiculars to MN. If we suppose that K, L are not distinct points, but one point, the axiom that two straight lines cannot enclose a space is not contradicted.] II. Ptolemy now tries to prove 1. 29 without using our Postulate, and then deduces the Postulate from it (Proclus, pp. 365, 14 — 367, 37). The argument to prove 1. 29 is as follows. The straight line which cuts the parallels must make the sum of the interior angles on the same side equal to, greater than, or less than, two right angles. b £ P "Let AB, CD be parallel, and let FG meet them. I say (1) that FG does not make the interior angles on the same side greater than two £— right angles. " For, if the angles AFG, CGF are greater than two right angles, the remaining angles BFG, DGF are less than two right angles. " But the same two angles are also greater than two right angles ; for AF, CG are no more parallel than FB, GD, so that, if the straight line falling on AF, CG makes the interior angles greater than two right angles, the straight line falling on FB, GD will also make the interior angles greater than two right angles. " But the same angles are also less than two right angles ; for the four angles AFG, CGF, BFG, DGF are equal to four right angles : which fa impossible *' Similarly (2) we can show that the straight line falling on the parallels does not make the interior angles on the same side less than two right angles. " But (3), if it makes them neither greater nor less than two right angles, it can only make the interior angles on the same side equal to two right angles." III, Ptolemy deduces Post 5 thus : Suppose that the straight lines making angles with a transversa) less than two right angles do not meet on the side on which those angles are. Then, a fortiori, they will not meet on the other side on which are the angles greater than two right angles. Hence /he straight lines will not meet in either direction ; they are there- fore parallel. But, if so, the angles made by them with the transversal are equal to two right angles, by the preceding proposition (= 1. 29). Therefore the same angles will be both equal to and less than two right angles: which is impossible. Hence the straight lines will meet. ao6 BOOK I [i. Post. 5 IV. Ptolemy lastly enforces his conclusion that the straight lines will meet on the side on which are the angles less than two right angles by recurring to the a fortiori step in the foregoing proof. Let the angles AFG, CGF in the accompanying figure be together less than two right angles. Therefore the angles BFG, DGF are greater than two right angles. We have proved that the straight lines are not non-secant. If they meet, they must meet either towards A, C, or towards B, D. (i) Suppose they meet towards B, £>, at K. Then, since the angles AFG, CGFsze. less than two right angles, and the angles AFG, GFB are equal to two right angles, take away the common angle AFG, and the angle CGF is less than the angle BFG; that is, the exterior angle of the triangle KFG is less than the interior arid opposite angle BFG : which is impossible. Therefore AB, CD do not meet towards B, D. (v) But they do meet, and therefore they must meet in one direction or the other: therefore they meet towards A, B, that is, on the side where are the angles less than two right angles. The flaw in Ptolemy's argument is of course in the part of his proof of i. 2 9 which I have italicised. As Proclus says, he is not entitled to assume that, if AB, CD are parallel, whatever is true of the interior angles on one side of FG (i.e. that they are together equal to, greater than, or less than, two right angles} is necessarily true at the same time of the interior angles on the other side. Ptolemy justifies this by saying that FA, GC are no more parallel in one direction than FB, GD are in the other : which is equivalent to the assumption that through any point only one parallel can be drawn to a given straight line. That is, he assumes an equivalent of the very Postulate he is endeavouring to prove. Proclus. Before passing to his own attempt at a proof, Proclus (p. 368, 26 sqq.) examines an ingenious argument (recalling somewhat the famous one about Achilles and the tortoise) which appeared to show that it was impossible for the lines described in the Postulate to meet. Let AB, CD make with AC the angles BAC, ACD together less than two right angles. Bisect AC a.t E and along AB, CD respectively measure AF, CG so that each is equal to AE. gl \ |h Bisect FG at K and mark off FK, GL each equal to FH; and so on. Then AF, CG will not meet at any point on FG ; for, if that were the case, two sides of a triangle would be together equal to the third : which is impossible. 1. Post. 5] NOTE ON POSTULATE 5 *>7 Similarly, AB, CD will not meet at any point on KL; and "proceeding like this indefinitely, joining the non-coincident points, bisecting the lines so drawn, and cutting off from the straight lines portions equal to the half of these, they say they thereby prove that the straight lines AB, CD will not meet anywhere." It is not surprising that Proclus does not succeed in exposing the fallacy here (the fact being thai the process will indeed be endless, and yet the straight lines will intersect within a finite distance). But Proclus' criticism contains nevertheless something of value. He says that the argument will prove too much, since we have only to join A G in order to see that straight lines making svme angles which are together less than two right angles do in fact meet, namely AG, CG. "Therefore it is not possible to assert, without some definite limitation, that the straight lines produced from angles less than two right angles do not meet. On the contrary, it is manifest that MM straight lines, when produced from angles less than two right angles, do meet, although the argument seems to require it to be proved that this property belongs to all such straight lines. For one might say that, the lessening of the two right angles being subject to no limitation, with such and such an amount of lessening the straight tines remain non-secant, but with an amount of lessening in excess of this they meet (p. 371, 2 — 10)." [Here then we have the germ of such an idea as that worked out by Lobachewsky, namely that the straight lines issuing from a point in a plane can be divided with reference to a straight line lying in that plane into two classes, "secant" and "non-secant," and that we may define as parallel the two straight lines which divide the secant from the non-secant class.] Proclus goes on (p. 371, io) to base his own argument upon "an axiom such as Aristotle too used in arguing that the universe is finite. For, if from one point two straight lines forming an angle be produced indefinitely, the distance (StaiTTairit, Arist. htaimjjta) between the said straight tines produced indefinitely will exceed any finite magnitude. Aristotle at all events showed that, if the Straight lines drawn from the centre to the circumference are infinite, the interval between them is infinite. For, if it is finite, it is impossible to increase the distance, so that the straight lines {the radii) are not infinite. Hence the straight lines, when produced indefinitely, will be at a distance from one another greater than any assumed finite magnitude." This is a fair representation of Aristotle's argument in De caelo ]. 5, 271 b 28, although of course it is not a proof of what Proclus assumes as an axiom. This being premised, Proclus proceeds (p. 371, 24): I. " I say that, if any straight line cuts one of two parallels, it will cut the other also. "For let AB, CD be parallel, and let EFG cut AB; I say that it will cut CD also. " For, since BF, FG are two straight lines from .E one point F, they have, when produced indefinitely, ft X^ B a distance greater than any magnitude, so that it will ^\ also be greater than the interval between the parallels. " Whenever therefore they are at a distance from one another greater than the distance between the parallels, FG will cut CD. " Therefore etc." 2o8 BOOK I [i. Post. 5 II. " Having proved this, we shall prove, as a deduction from it, the theorem in question. "For let AB, CD be two straight lines, and let EF falling on them make the angles BEF, DFE less than two right angles. "I say that the straight lines will meet on that side on which are the angles less than two right angles. " For, since the angles BEF, DFE are less than two right angles, let the angle HEB be equal to the excess of two right angles (over them), and let HE be produced to K. " Since then EF falls on KH, CD and makes the two interior angles HEF, DFE equal to two right angles, the straight lines MX, CD are parallel. "And AB cuts KH\ therefore it will also cut CD, by what was before shown. " Therefore AB, CD will meet on that side on which are the angles less than two right angles. " Hence the theorem is proved." Clavius criticised this proof on the ground that the axiom from which it starts, taken from Aristotle, itself requires proof. He points out that, just as you cannot assume that two lines which continually approach one another will meet (witness the hyperbola and its asymptote), so you cannot assume that two lines which continually diverge will ultimately be so far apart that a perpendicular from a point on one let fall on the other will be greater than any assigned distance ; and he refers to the conchoid of Nkomedes, which continually approaches its asymptote, and therefore continually gets farther away from the tangent at the vertex • yet the perpendicular from any point on the curve to that tangent will always be less than the distance between the tangent and the asymptote. Saccheri supports the objection. Proclus' first proposition is open to the objection that it assumes that two "parallels" (in the Euclidean sense) or, as we may say, two straight liius which have a common perpendicular, are (not necessarily equidistant, but) so related that, when they are produced indefinitely, the perpendicular from a point of one upon the other remains finite. This last assumption is incorrect on the hyperbolic hypothesis ; the "axiom" taken from Aristotle does not hold on the elliptic hypothesis, Nasiraddln at.-TusI. The Persian-bom editor of Euclid, whose date is 1201— 1274, has three lemmas leading up to the final proposition. Their content is substantially as follows, the first lemma being apparently assumed as evident I. (o) If AB, CD be two straight lines such that successive perpen- diculars, as EF, GH, KL, from points on AB to CD always make with A3 unequal angles, which are always acute on the side towards B and always obtuse on the side towards A, then the lines AB, CD, so long as they do not cut, approach continually nearer in the direction of the acute angles and diverge continually in the direction of the obtuse angles, and the perpendiculars diminish towards B, D, and in- crease towards A, C. (6) Conversely, if the perpendiculars so drawn L H F C continually become shorter in the direction of B, D, and longer in the i. Post. 3] NOTE ON POSTULATE 5 zog direction of A, C, the straight lines AB, CD approach continually nearer in the direction of B, D and diverge continually in the other direction ; also each perpendicular will make with AB two angles one of which is acute and the other is obtuse, and all the acute angles will lie in the direction towards B, D, and the obtuse angles in the opposite direction. [Saccheri points out that even the first part {a) requires proof. As regards the converse (b) he asks, why should not the successive acute angles made by the perpendiculars with AB, while remaining acute, become greater and greater as the perpendiculars become smaller until we arrive at last at a perpendicular which is a common perpendicular to both lines? If that happens, alt the author's efforts are in vain. And, if you are to assume the truth of the statement in the lemma without proof, would it not, as Wall is said, be as easy to assume as axiomatic the statement in Post. 5 without more ado?] II. ^AC, BI) be drawn from the extremities of AB at right angles to it and on the same side, and if AC, EDfc made equal to one another and CD be joined, each of the angles ACD, BDC will be right, and CD will be equal to AB, The first part of this lemma is proved by redwtio ad absurdum from the preceding lemma. If, e.g., the angle A CD is not right, it must either be acute or obtuse. Suppose it is acute ; then, by lemma 1, A C is greater than BD, which '\s contrary to the hypothesis. And so on. The angles ACD, BDC being proved to be right angles, it is easy to prove that AB, CD are equal. [It is of course assumed in this " proof" that, if the angle ACD is acute, the angle BDC is obtuse, and vice versa.] III. /// any triangle the three angles are together equal to two right angles. This is proved for a right-angled triangle by means of the foregoing lemma, the four angles of the quadrilateral ABCD of that lemma being all right angles. The proposition is then true for any triangle, since any triangle can be divided into two right-angled triangles IV. Here we have the final " proof " of Post. 5. Three cases are distinguished, 'but it is enough to show the case where one of the interior angles is right and the other acute. Suppose AB, CD to be two straight lines met by FCE making the angle ECD a right angle and the angle CEB an acute angle. 'lake any point G on EB, and draw GH perpendicular to EC. Since the angle CEG is acute, the perpendicular GH will fall on the side of E towards D, and will either coincide with CD or not coincide with it. In the former case the proposition is proved. If GH does not coincide with CD but falls on the side of it towards F, CD, being within the triangle formed by the perpendicular and by CE, EG, must cut EG. [An axiom is here used, namely that, if CD be produced far enough, it must pass outside the triangle and therefore cut some side, which must be EB, since it cannot be the perpendicular (1. 27), or CE.) Lasdy, let C.fffall on the side of CD towards E. aro BOOK I [i. Post. 5 Along HC set off HK, KL etc., each equal to EH, until we get the first point of division, as M, beyond C. Along GB set off GN, NO etc, each equal to EG, until EP is the same multiple of EG that EM is of EH. Then we can prove that the perpendiculars from N, 0, P on EC fall on the points K, L, M respectively. For take the first perpendicular, that from N, and call it NS. Draw EQaa right angles to EH and equal to GH, and set off SR along SiValso equal to GH, Join QG, GR, Then (second lemma) the angles EQG, QGHaie right, and QG = EH. Similarly the angles SRG, RGHwe right, and RG-SH Thus RGQ is one straight line, and the vertically opposite angles NGR, EGQ are equal. The angles NRG, EQG are both right, and NG = GE, by construction. Therefore (1. 26") EG =■ GQ ; whence SH= HE = KH, and S coincides with K. We may proceed similarly with the other perpendiculars. Thus PM is perpendicular to EE. Hence CD, being parallel to MP and within the triangle PME, must cut EP, if produced far enough. John Wallis. As is well known, the argument of Wallis (1616—1703) assumed as a postulate that, given a figure, another figure is possible which is similar to (he given one and of any sine whatever. In fact Wallis assumed this for triangles only. He first proved (1) that, if a finite straight line is placed on an infinite straight line, and is then moved in its own direction as far as we please, it will always lie on the same infinite straight line, (2) that, if an angle be moved so that one leg always slides along an infinite straight line, the angle will remain the same, or equal, (3) that, if two straight lines, cut by a third, make the interior angles on the same side less than two right angles, each of the exterior angles is greater than the opposite interior angle (proved by means of 1. 13). p x (4) UAB, CD make, with AC, the interior angles less than two right angles, suppose AC (with AB rigidly attached to it) to move along _ ^ \- g AF to the position ay, such that a coincides with C. If AB then takes the position aft, o£ lies entirely outside CD (proved by means of {3) above). (5) With the same hypotheses, the straight line aft, sr AB, during Us motion, and before a reaches C, must cut the straight tine CD. S6) Here is enunciated the postulate stated above. 7-) Postulate S is now proved thus. Let AB, CD be the straight lines which make, with the infinite straight line ACF meeting them, the interior angles BA C, DC A together less than two right angles. Suppose AC (with AB rigidly attached to it) to move along ACF until AB takes the position of aft cutting CD in it. Then, «CVr being a triangle, we can, by the above postulate, suppose a triangle drawn on the base CA similar to the triangle aCVr. Let it be ACF. [Wallis here interposes a defence of the hypothetical construction.] 1. Post. 5] NOTE ON POSTULATE 5 sit Thus CP and AP meet at P; and, as by the definition of similar figures the angles of the triangles PCA, rCa are respectively equal, the angle PCA being equal to the angle rCa and the angle PAC to the angle miCor BAC, it follows that CP, APMe on CD, A3 produced respectively. Hence AB, CD meet on the side on which are the angles less than two right angles. [The whole gist of this proof lies in the assumed postulate as to the existence of similar figures ; and, as Saccheri points out, this is equivalent to unconditionally assuming the "hypothesis of the right angle," and consequently Euclid's Postulate 5.] Gerolamo Saccheri. The book Euclides ab omni naevo vindicatus (1733) by GeTolamo Saccheri {1667 — 1733), a Jesuit, and professor at the University of Pavia, is now accessible (1) edited in German by Engel and Stackel, Die Theorie dtr Parallellinien von Euhtid bis auf Gauss, 1895, pp. 41 — 136, and (2) in an Italian version, abridged but annotated, L'Euclide emendato del P. Gerolamo Saccheri, by G. Boccardini (Hoepli, Milan, 1904}. It is of much greater importance than all the earlier attempts to prove Post. 5 because Saccheri was the first to contemplate the possibility of hypotheses other than that of Euclid, and to work out a number of consequences of those hypotheses. He was therefore a true precursor of Legendie and of Lobachewsky, as Beltrami called him (1889), and, it might be added, of Riemann also. For, as Veronese observes {Fondamenti di geometria, p, 570), Saccheri obtained a glimpse of the theory of parallels in all its generality, while Legendre, Lobachewsky and G, Bolyai excluded a priori, without knowing it, the "hypo- thesis of the obtuse angle," or the Riemann hypothesis. Saccheri, however, was the victim of the preconceived notion of his time that the sole possible geometry was the Euclidean, and he presents the curious spectacle of a man laboriously erecting a structure upon new foundations for the very purpose of demolishing it afterwards ; he sought for contradictions in the heart of the systems which he constructed, in order to prove thereby the falsity of his hypotheses. For the purpose of formulating his hypotheses he takes a plane quadri- lateral ABDC, two opposite sides of which, A C, BD, are equal and perpendicular to a third AB. Then the angles at C and D are easily proved to be equal. On the Euclidean hypothesis they are both right angles; but apart from this hypothesis they might be both obtuse or both acute. To the three possibilities, whicfc Saccheri distinguishes by the names (1) the hypothesis of the right angle, (i) the hypothesis of the obtuse angle and (3) the hypothesis of the acute angle respectively, there corresponds a certain group of theorems ; and Saccheri's point of view is that the Postulate will be completely proved if the consequences which follow from the last two hypotheses comprise results inconsistent with one another. Among the most important of his propositions are the following : (1) If the hypothesis of the right angle, or of the obtuse angle, or of the acute angle is proved true in a single case, it is true in every other case. (Props, v., VI., VII.) (2) According as the hypothesis of the right angle, the obtuse angle, or the acute angle is true, the sum of the thru angles of a triangle is equal to, greater than, or less than two right angles. (Prop. i\. ) lit BOOK I [i. Post. 5 (3) From the existence of a single triangle in which the sum of the angles is equal to, greater than, or less than two right angles (lie truth of the hypothesis of the right angle, obtuse angle, or acute angle respectively follows. (Prop, xv.) These propositions involve the following : If in a single triangle the sum of the angles is equal to, greater than, or less than two right angles, then any triangle has the sum of its angles equal to, greater than, or less than tlt'O right angles respectively, which was proved about a century later by Legendre for the two cases only where the sum is equal to or less than two right angles. The proofs are not free from imperfections, as when, in the proofs of Prop. xii. and the part of Prop. xm. relating to the hypothesis of the obtuse angle, Saccheri uses Eucl. 1. 18 depending on 1. 16, a proposition which is only valid on the assumption that straight lines are infinite in length ; for this assumption itself does not hold under the hypothesis of the obtuse angle (the Riemann hypothesis). The hypothesis of the acute angle takes Saccheri much longer to dispose of, and this part of the book is less satisfactory ; but it contains the following propositions afterwards established anew by Lobachewsky and Bolyai, viz. : (4) Two straight lines in a plane {even on the hypothesis of the acute angle) either have a common perpendicular, or must, if produced in one and the same direction, either intersect once at a finite distance or at least continually approach one another. {Prop, xxin.) (5) In a cluster of rays issuing from a point there exist always (on the hypothesis of the acute angle) two determinate straight lines which separate the straight lines which intersect a fixed straight line from those which do not intersect it, ending with and including the straight line which has a common perpendicular with the fixed straight line. (Props. XXX., xxxc, xxxii.) Lambert. A dissertation by G.S. Kliigel, Conatuum praecipuorum tlteoriamparallelarnm demonstrandi recensio (1 763), contained an examination of some thirty " demon- strations" of Post. 5 and is remarkable for its conclusion expressing, apparently for the first time, doubt as to its demenstrability and observing that the certainty which we have in us of the truth of the Euclidean hypothesis is not the result of a series of rigorous deductions but rather of experimental observations. It also had the greater merit that it called the attention of Johann Heinrich Lambert (1728—1777) to the theory of parallels. His Theory of Parallels was written in 1766 and published after his death by G. Bernoulli and C. F. Hindenburg ; it is reproduced by Engel and Stackel (op. sit. pp. 151 — 208). The third part of Lambert's tract is devoted to the discussion of the same three hypotheses as Saccheri's, the hypothesis of the right angle being for Lambert the first, that of the obtuse angle the second, and that of the acute angle the third, hypothesis; and, with reference to a quadrilateral with three right angles from which Lambert starts (that is, one of the halves into which the median divides Saccheri's quadrilateral), the three hypotheses are the assumptions that the fourth angle is a right angle, an obtuse angle, or an acute angle respectively. Lambert goes much further than Saccheri in the deduction of new propositions from the second and third hypotheses. The most remarkable is the following. The area of a plane triangle, under the second and third hypotheses, is proportional to the difference between the sum vf the three angles and two right angles. h Post. 5] NOTE ON POSTULATE 5 «3 Thus the numerical expression for the area of a triangle is, under the third hypothesis &.=;A( r -A-£-C) (1), and under the second hypothesis &. = *(A + £+C-*) (a), where A is a positive constant A remarkable observation is appended (5 82) : "In connexion with this it seems to be remarkable that the second hypothesis holds if spherical instead of plane triangles are taken, because in the former also the sum of the angles is greater than two right angles, and the excess is proportional to the area of the triangle. " It appears still more remarkable that what I here assert of spherical triangles can be proved independently of the difficulty of parallels. This discovery that the second hypothesis is realised on the surface of a sphere is important in view of the development, later, of the Riemann hypothesis (1854). Still more remarkable is the following prophetic sentence : " I am almost inclined to draw the conclusion that the third hypothesis arises with an imaginary spherical surface" (cf. Lobachewsky's Gcome'trie imaginaire, 1837). No doubt Lambert was confirmed in this by the fact that, in the formula (1) above, which, for h = r 1 , represents the area of a spherical triangle, if r V- 1 is substituted for r, and r 1 = k, we obtain the formula (1). Legend re. No account of our present subject would be complete without a full reference to what is of permanent value in the investigations of Adrien Marie Legendre {1752 — 1833) relating to the theory of parallels, which extended over the space of a generation. His different attempts to prove the Euclidean hypothesis appeared in the successive editions of his aliments de Giomklrie from the first {1794) to the twelfth {1823), which last may be said to contain his last word on the subject. Later, in 1833, he published, in the Afhnoires de I'Acadimie Royals des Sciences, xn. p. 367 sqq., a collection of his different proofs under the title Reflexions sur dffirentes maniires de dhnontrer la thiorie des paralteles. His exposition brought out clearly, as Saccheri had done, and kept steadily in view, the essential connexion between the theory of parallels and the sum of the angles of a triangle. In the first edition of the Elements the proposition that the sum of the angles of a triangle is equal to two right angles was proved analytically on the basis of the assumption that the choice of a unit of length does not affect the correctness of the proposition to be proved, which is of course equivalent to Wallis' assumption of the existence of similar figures. A similar analytical proof is given in thj notes to the twelfth edition. In his second edition Legendre proved Postulate 5 by means of the assumption that, given three points not in a straight line, there exists a circle passing through all three. In the third edition (1800) he gave the proposition that the sum of the angles of a triangle is not greater than two right angles ; this proof, which was geometrical, was replaced later by another, the best known, depending on a construction like that of Euclid 1. 16, the continued application of which enables any number of successive triangles to be evolved in which, while the sum of the angles in each remains always equal to the sum of the angles of the original triangle, one of the angles increases and the sum of the other two diminishes continually. But Legendre found the proof of the equally necessary proposition that the sum of the angles of a triangle is ai 4 BOOK I [i. Post, S not less than two right angles to present great difficulties. He first observed that, as in the case of spherical triangles (in which the sum of the angles is greater than two right angles) the excess of the sura of the angles over two right angles is proportional to the area of the triangle, so in the case of rectilineal triangles, if the sum of the angles is less than two right angles by a Mrtain deficit, the deficit will be proportional to the area of the triangle. Hence if, starting from a given triangle, we could construct another triangle in which the original triangle is contained at least m times, the deficit of this new triangle will be equal to at least m times that of the original triangle, so that the sum of the angles of the greater triangle will diminish progressively as m increases, until it becomes zero or negative : which is absurd. The whole difficulty was thus reduced to that of the construction of a triangle containing the given triangle at least twice ; but the solution of even this simple problem requires it to be assumed (or proved) that through a given point within a given angle less than two-thirds of a right angle we can always draw a straight tine which shall meet both sides of the angle. This is however really equivalent to Euclid's Postulate. The proof in the course of which the necessity for the assumption appeared is as follows. It is required to prove that the sum of the angles of a triangle cannot be less than two right angles. Suppose A is the least of the three angles of a triangle ABC. Apply to the opposite side 2?C a triangle DBC, equal to the triangle ACB, and such that the angle DBC is equal to the angle ACB, and the angle DCB to the angle ABC ; and draw any straight line through D cutting AB, AC produced in E, F. If now the sum of the angles of the triangle ABC is less than two right angles, being equal to aB-i say, the sum of the angles of the triangle DBC, equal to the triangte ABC, is also 2^-8. Since the sum of the three angles of the remaining triangles DEB, FDC respectively cannot at all events be greater than two right angles [for I>egendre's proofs of this see below], the sum of the twelve angles of the four triangles in the figure cannot be greater than 4B + {2B - &) + (2B - ty, i.e. %R-al. Now the sum of the three angles at each of the points B, C, D is iR. Subtracting these nine angles, we have the result that the three angles of the triangle AEF cannot be greater than 2R - 28. Hence, if the sum of the angles of the triangle ABC is less than two right angles by £, the sum of the angles of the- larger triangle AEF is less than two right angles by at least 28. We can continue the construction, making a still larger triangle from AEF, and so on. But, however small 8 is, we can arrive at a multiple 2*$ which shall exceed any given angle and therefore tR itself; so that the sum of the three angles of a triangle sufficiently large would be zero or even less than zero : which is absurd. Therefore etc. The difficulty caused by the necessity of making the above-mentioned assumption made Legendre abandon, in his ninth edition, the method of the i. Post. 5] NOTE ON POSTULATE 5 115 editions from the third to the eighth and return to Euclid's method pure and simple. But again, in the twelfth, he returned to the plan of constructing any number of successive triangles such that the sum of the three angles in all of them remains equal to the sum of the three angles of the original triangle, but two of the angles of the new triangles become smaller and smaller, while the third becomes larger and larger ; and this time he claims to prove in one proposition that the sum of the three angles of the original triangle is equal to two right angles by continuing the construction of new triangles indefinitely and compressing the two smaller angles of the ultimate triangle into nothing, while the third angle is made to become a flat angle at the same time. The construction and attempted proof are as follows. Let ABC be the given triangle ; let A B be the greatest side and BC the least ; therefore C is the greatest angle and A the least. From A draw AD to the middle point of BC, and produce AD to C, making AC equal to AB, Produce AB to B 1 , making AB equal to twice AD. The triangle ABC is then such that the sum of its three angles is equal to the sum of the three angles of the triangle ABC. For take AK along AB equal to AD, and join C'K. Then the triangles ABD, ACK havt two sides and the included angles respectively equal, and are therefore equal in all respects ; and C'K is equal to BD or DC. Next, in the triangles BCK, A CD, the angles BKC, ADC are equal, being respectively supplementary to the equal angles AKC, ADB; and the two sides about the equal angles are respectively equal ; therefore the triangles BC'K, A CD are equal in all respects. Thus the angle AC'B is the sum of two angles respectively equal to the angles B, C of the original triangle ; and the angle A in the original triangle is the sum of two angles respectively equal to the angles at A and B' in the triangle ABC. It follows that the sum of the three angles of the new triangle ABC is equal to the sum of the angles of the triangle ABC. Moreover, the side AC, being equal to AB, and therefore greater than AC, is greater than BC which is equal to AC. Hence the angle C'AB'w less than the angle ABC ; so that the angle CAB is less than \A, where A denotes the angle CAB of the original triangle. [It will be observed that the triangle ABC is really the same triangle as the triangle ABB obtained by the construction of Eucl. 1. 16, but differently placed so that the longest side lies along AB.] By taking the middle point D of the side BC and repeating the same construction, we obtain a triangle AB'C" such that (1) the sum of its three angles is equal to the sum of the three angles of ABC, (a) the sum of the ai6 BOOK I [1. Post. 5 two angles CAB", AB"C" is equal to the angle CAB in the preceding triangle, and is therefore less than \A, and (3) the angle CAB' is less than half the angle CAB, and therefore less than \A. Continuing in this way, we shall obtain a triangle Abe such that the sum of two angles, those at A and i, is less than — A, and the angle at c is greater than the corresponding angle in the preceding triangle. If, Legendre argues, the construction be continued indefinitely so that - n A becomes smaller than any assigned angle, the point c ultimately ties on Alt, and the sum of the three angles of the triangle (which is equal to the sum of the three angles of the original triangle) becomes identical with the angle at c, which is then a.Jlat angle, and therefore equal to two right angles. This proof was however shown to be unsound (in respect of the final inference) by J. P. W. Stein in Gergonne's Annaln de Mathimatiques XV., 1824, pp. 77—79. We will now reproduce shortly the substance of the theorems of Legendre which are of the most permanent value as not depending on a particular hypothesis as regards parallels. I. The sum of the three angles of a triangle cannot it greater than two right angles. This Legendre proved in two ways. (r) Mr si proof (in the third edition of the Aliments). Let ABC be the given triangle, and ACf a straight line. Make CE equal to AC, the angle DCE equal to the angle BAC, and DC equal to AB. Join DE. Then the triangle DCE is equal to the triangle BAC in all respects. If then the sum of the three angles of the triangle ABC is greater than %R, the said sum must be greater than the sum of the angles BCA, BCD, DCE, which sum is equal to 2H. Subtracting the equal angles on both sides, we have the result that the angle ABC is greater than the angle BCD. But the two sides AB, BC of the triangle ABC are respectively equal to the two sides DC, CB of the triangle BCD. Therefore the base AC is greater than the base BD (Eucl. 1. 14). Next, make the triangle BEG {by the same construction) equal in all respects to the triangle BAC or DCE ; and we prove in the same way that CE (or AC) is greater than DE. And, at the same time, BD is equal to DE, because the angles BCD, DEE are equal. Continuing the construction of further triangles, however small the difference between AC and BD is, we shall ultimately reach some multiple i. Post. 5] NOTE ON POSTULATE 5 ai7 of this difference, represented in the figure by (say) the difference between the straight line AJ and the composite line BDFHK, which will be greater than any assigned length, and greater therefore than the sum of AB and JJC. Hence, on the assumption that the sum of the angles of the triangle ABC is greater than 2R, the broken line ABDFHKJ may be less than the straight Hne AJ: which is impossible. Therefore etc. (2) Proof substituted later. If possible, let 2^ + a be the sum of the three angles of the triangle ABC, of which A is not greater than either of the others. Bisect BC at H, and produce AH to D, making HD equal to AH ; join BD. Then the triangles AHC, DHB are equal in all respects (l. 4); and the angles CAH,ACHaie respectively equal to the angles BDH, DBH. It follows that the sum of the angles of the triangle ABD is equal to the sum of the angles of the original triangle, i.e. to tR + a. And one of the angles DAB, ADB is either equal to or less than half the angle. CAB. Continuing the same construction with the triangle ADB, we find a third triangle in which the sum of the angles is still zR + a, while one of them is equal to or less than J L CAB)jj\. Proceeding in this way, we arrive at a triangle in which the sum of the angles is 2R + a, and one of them is not greater than ( L CAB)J2 H . And, if n is sufficiently large, this will be less than a. ; in which case we should have a triangle in which two angles are together greater than two right angles : which is absurd. Therefore a is equal to or less than zero. (It will be noted that in both these proofs, as in Eucl. 1. 16," it is taken for granted that a straight line is infinite in length and does not return into itself, which is not true under the Riemann hypothesis.) II. On the assumption that the sum of the angles of a triangle is less than two right angles, if a triangle is made up of two others, the " deficit" of ' tht, former is equal to the sum of the " deficits " of the others. In fact, if the sums of the angles of the component triangles are 2R -a, 2R - fi respectively, the sum of the angles of the whole triangle is (zj¥-a) + (2JP-£)-3.ff = 7Je-fa + P). III. If the sum of the three angles of a triangle is equal to two right angles, the same is true of all triangles obtained by subdividing it by straight lines drawn from a vertex to meet the opposite side. Since the sum of the angles of the triangle ABC is equal to 2R, if the sum of the angles of the triangle ABD were 2R - a, it would follow that the sum of the angles of the triangle A ADC must be »R + q, which is absurd (by I. above). IV. If in a triangle the sum of the three angles is equal to two right angles, a quadrilateral can always be constructed with four right angles and four equal sides B^ £2 io exceeding in length any assigned rectilineal segment. Let ABC be a triangle in which the sum of the angles is equal to two «i8 BOOK I [i. Post. 5 right angles. We can assume ABC to be an isosceles right-angled triangle because we can reduce the case to this by making subdivisions of ABC by straight lines through vertices (as in Prop. III. above). Taking two equal triangles of this kind and placing their hypotenuses together, we obtain a quadrilateral with four right angles and four equal sides. Putting four of these quadrilaterals together, we obtain a new quadrilateral 0/ the same kind but with its sides double of those of the first quadrilateral. After n such operations we have a quadrilateral with four right angles and four equal sides, each being equal to 3" times the side AH. The diagonal of this quadrilateral divides it into two equal isosceles right- angled triangles in each of which the sum of the angles is equal to two right angles. Consequently, from the existence ot one triangle in which the sum of the three angles is equal to two right angles it follows that there exists an isosceles right-angled triangle with sides greater than any assigned rectilineal segment and such that the sum of its three angles is also equal to two right angles. V. If the sum of the three angles of one triangle is equal to two right angles, the sum of the three angles of any other triangle is also equal to two right angles. It is enough to prove this for a right-angled triangle, since any triangle can be divided into two right-angled triangles. Let ABC be any right-angled triangle. If then the sum of the angles of any one triangle is equal to two right angles, we can construct (by the preceding Prop.) an isosceles right-angled triangle with the same property and with its perpendicular sides, greater than those of ABC. Let A'BC' be such a triangle, and let it be applied to ABC, as in the figure. Applying then Prop. 111. above, we deduce first that the sum of the three angles of the triangle ABC is equal to two right angles, and next, for the same reason, that the sum of the three angles of the original triangle A BC is equal to two right angles. VI. If in any one triangle the sum of the three angles is less than two right angles, the sum of the three angles of any other triangle is also less than two right angles. This follows from the preceding theorem. (It will be observed that the last two theorems are included among those of Saccheri, which contain however in addition the corresponding theorem touching the case where the sum of the angles is greater than two right angles.) We come now to the bearing of these propositions upon Euclid's Postulate 5 ; and the next theorem is VII. If the sum of the three angles of a triangle is equal to two right angles, through any point in a plane there (an only be drawn one parallel to a given straight line. r. Post. 5] NOTE ON POSTULATE 5 219 For the proof of this we require the following Lemma. // is always possible, through a faint P, to draw a straight line which shall make, with a ghten straight line (r), an angle less than any assigned angle. Let Q be the foot of the perpendicular from /'upon r. Let a segment QR be taken on r, on either side of Q, such that QR is equal to PQ. Join PR, and mark off the segment RR' equal to PR ; join PR'. If tu represents the angle QPR or the angle QRP, each of the equal angles RPR', RR'P is not greater than 10/ 1. Continuing the construction, we obtain, after the requisite number of operations, a triangle PR,-, R n in which each of the equal angles is equal to or less than <o/*". Hence we shall arrive at a straight line PR, which, starting from /'and meeting r, makes with r an angle as small as we please. To return now to the Proposition. Draw from P the straight line s perpendicular to PQ. Then any straight line drawn from P which meets r in R will form equal angles with r and s, since, by hypothesis, the sum of the angles of the triangle PQR is equal to two right angles. And since, by the Lemma, it is always possible to draw through P straight lines which form with r angles as small as we please, it follows that all the straight lines through /', except s, will meet r. Hence s is the only parallel to r that can be drawn through P. The history of the attempts to prove Postulate 5 or something equivalent has now been brought down to the parting of the ways. The further developments on lines independent of the Postulate, beginning with Schweikart {1780 — 1857), Taurinus {1794 — 1874), Gauss {1777 — 1855), Lobacbewsky {1793 — 1850), J. Bolyai (180* — 1860), Riemann (1826— 1866), belong to the history of non-Euclidean geometry, which is outside the scope of this work. I may refer the reader to the full article Sulla teoria delle par allele e suite geometric non-euelidec by R. Bonola in Questioni riguardanti le matematiche elementari, I., of which I have made considerable use in the above, to the same author's La geometria non-euclidea, Bologna, 1906, to the first volume of Killing's Einfuhrutig in die Grundlagen der Geometric, Paderborn, 1893, to P. Mansion's Premiers princifes de mitag^omitrie, and P. Barbarin's La giomitrie ncn-Euclidicnne, Paris, 1902, to the historical summary in Veronese's Fondamenti di geometria, 1891, p. 565 sqq., and (for original sources) to Engel and Stackel, Die Theorie der Paralleltinicn von Eukltd bis auf Gauss, 1895, and Urkunden zur Gcschichte der nickt-Euklidischtn Geometric, 1. (Lobachewsky), 1899, and ti. (Wolfgang und Johann Bolyai). I will only add that it was Gauss who first expressed a conviction that the Postulate could never be proved ; he indicated this in reviews in the Gottin- gische gelckrfe Anzeigen, 20 Apr. 18 16 and 18 Oct. 1822, and affirmed it in a letter to Bessei of 27 January, 1829. The actual indemonstrability of the Pos- tulate was proved by Beltrami (1868) and by Houel {Note sur rimpessibilite" de dimontrer par une construction plane leprincipe de la thiorie des paralletes dit Pos- tulatum (f.EwA'AmBattaglini's Giornale di matematiehe,\ lit., 1870, pp.84 — 89). BOOK I [i. Post. 5 Alternatives for Postulate 5. It may be convenient to collect here a few of the more noteworthy substitutes which have from time to time been formally suggested or tacitly assumed. { i ) Through a given point only one parallel can be drawn to a given straight line or, Two straight lines which intersect one another cannot both be parallel to one and the same straight line. This is commonly known as " Playfair's Axiom," but it was of course not a new discovery. It is distinctly stated in Proclus' note to Eucl. I. 31. (r a) If a straight tine intersect one of two parallels, it will intersect the other also (Proclus). (1 b) Straight lines parallel to the same straight line are parallel to one another. The forms (1 a) and (1 b) are exactly equivalent to (i). {2) There exist straight lines everywhere equidistant from one another (Posidonius and Geminus) ; with which may be compared Proclus' tacit assumption that Parallels remain, throughout their length, at a finite distance from one another. (3) There exists a triangle in which the sum of the three angles is equal to two right angles (Legendre). (4) Given any figure, there exists a figure similar to it of any size toe please (Wallis, Carnot, Laplace). Saccheri points out that it is not necessary to assume so much, and that it is enough to postulate that there exist two unequal triangles with equal angles. (5) Through any point within an angle less than two-thirds of a right angle a straight tine can always be drawn which meets both sides of the angle {Legendre). With this may be compared the similar axiom of Lorenz (Grundriss der r einen und angewandten Mathematik, 1791): Every straight line through a point within an angle must meet one of the sides of the angle. (6) Given any three points not in a straight line, there exists a circle passing through them {Legendre, W, Bolyai). (7) " If I could prove that a rectilineal triangle is possible the content of which is greater than any given area, I am in a position to prove perfectly rigorously the whole of geometry" (Gauss, in a letter to W. Bolyai, 1 799). Cf. the proposition of Legendre numbered iv. above, and the axiom of Worpitzky: There exists no triangle in which every angle is as small as we please. (8) If in a quadrilateral three angles are right angles, the fourth angle is a right angle also (Clairaut, 1741). (9) If two straight lines are parallel, they arc figures opposite to (or the reflex of) one another with respect to the middle points of ail their transversal segments (Veronese, Elements, 1904). Or, Two parallel straight lines intercept, on every transiiersal which passes through the middle point of a segment included between them, another segment the middle point of which is the middle point of the first (Ingrami, Etementi, 1904). Veronese and Ingrami deduce immediately Playfair's Axiom, NOTES ON THE COMMON NOTIONS 221 AXIOMS OR COMMON NOTIONS. In a paper Sur Pauthenticite des axiomes d'Euclide in the Bulletin des sciences math, tt astron. 1884, p. i6z sq. {Mimoires scientifigues, ir., pp, 48—63), Paul Tannery maintained that the Common Notions (including the first three) were not in Euclid's work but were interpolated later. The following are his main arguments. (1) If Euclid had set about distinguishing between indemon- strable principles (a) common to all demonstrative sciences and (b) peculiar to geometry, he would, says Tannery, certainly not have placed the common principles second and the special principles (the Postulates) first. (2) If the Common Notions are Euclid's, this designation of them must be his too ; for he must have used some name to distinguish them from the Postulates and, if he had used another name, such as Axioms, it is impossible to imagine why that name was changed afterwards for a less suitable one. The word iwom {nation), says Tannery, never signified a notion in the sense of a proposition, but a notion of some object ; nor is it found in any technical sense in Plato and Aristotle. (3) Tannery's own view was that the formulation of the Common Notions dates from the time of Apollonius, and that it was inspired by his work relating to the Elements (we know from Proclus that Apollonius tried to prove the Common Notions), This idea, Tannery thought, was confirmed by a " fortunate coincidence " furnished by the occurrence of the word hrovt. {notion) in a quotation by Proclus (p. 100, 6): "we shall agree with Apollonius when he says that we have a notion {iwoaui) of a line when we order the lengths, only, of roads or walls to be measured." In reply to argument (r) that it is an unnatural order to place the purely geometrical Postulates first, and the Common Notions, which are not peculiar to geometry, last, it may be pointed out that it would surely have been a still more awkward arrangement to give the Definitions first and then to separate from them, by the interposition of the Common Notions, the Postulates, which are so closely connected with the Definitions in that they proceed to postulate the existence of certain of the things defined, namely straight lines and circles. (2) Though it is true that ii™« in Plato and Aristotle is generally a notion of an object, not of a fact or proposition, there are instances in Aristotle where it does mean a notion of a fact : thus in the Eth. Nic. ix. 11, 1 1 7 1 a 32 he speaks of " the notion (or consciousness) that friends sympathise " (1} iwom rod m>ya\yth Tok tt>£\uvs) and again, b 14, of "the notion (or consciousness) that they are pleased at his good fortune." It is true that Plato and Aristotle do not use the word in a technical sense ; but neither was there apparently in Aristotle's time any fixed technical term for what we call "axioms," since he speaks of them variously as "the so-called axioms in mathematics," "the so- called common axioms," " the common (things) " (to. jcotpd), and even " the common opinions " (kmvoi bofru). I see therefore no reason why Euclid should not himself have given a technical sense to " Common Notions," which is at least a distinct improvement upon " common opinions." (3) The use of fcroux in Proclus' quotation from Apollonius seems to me to be an unfortunate, rather than a fortunate, coincidence from Tannery's point of view, for it is there used precisely in the old sense of the notion of an object (in that case a line). No doubt it is difficult to feel certain that Euclid did himself use the term Common Notions, seeing that Proclus' commentary generally speaks of Axioms, But even Proclus (p. 194, 8), after explaining the meaning of the word "axiom," first as used by the Stoics, and secondly as used by "Aristotle and »** BOOK I [!. C. N. i the geometers," goes on to say : " For in their view (that of Aristotle and the geometers) axiom and common notion are the same thing." This, as it seems to me, may be a sort of apology for using the word " axiom " exclusively in what has gone before, as if Proclus had suddenly bethought himself that he had described both Aristotle and the geometers as using the one term " axiom," whereas he should have said that Aristotle spoke of " axioms," while "the geometers" (in fact Euclid), though meaning the same thing, called them Common Notions, It may be for a like reason that in another passage (p. 76, 16), after quoting Aristotle's view of an "axiom," as distinct from a postulate and a hypothesis, he proceeds : " For it is not by virtue of a common notion that, without being taught, we preconceive the circle to be such and such a figure." If this view of the two passages just quoted is correct, it would strengthen rather than weaken the case for the genuineness of Common Notions as the Euclidean term. Again, it is clear from Aristotle's allusions to the " common axioms in mathematics " that more than one axiom of this kind had a place in the text- books of his day ; and as he constantly quotes the particular axiom that, if equals be taken from equals, the remainders are equals which is Euclid's Common Notion 3, it would seem that at least the first three Common Notions were adopted by Euclid from earlier textbooks. It is ? besides, scarcely credible that, if the Common Notions which Apollonius tried to prove had not been introduced earlier (e.g. by Euclid), they would then have been interpolated as axioms and not as propositions to be proved. The line taken by Apollonius is much better explained on the assumption that he was directly attacking axioms which he found already admitted into the Elements. Proclus, who recognised the five Common Notions given in the text, warns us, not only against the error of unnecessarily multiplying the axioms, but against the contrary error of reducing their number unduly (p. 196, 15), "as Heron does in enunciating three only; for it is also an axiom that the whole is greater than the part, and indeed the geometer employs this in many places for his demonstrations, and again that things which coincide are equal." Thus Heron recognised the first three of the Common Notions ; and this fact, together with Aristotle's allusions to "common axioms" (in the plural), and in particular to our Common Notion 3, may satisfy us that at least the first three Common Notions were contained in the Elements as they left Euclid's hands. Common Notion i. Things which are equal to the same thing are also equal to one another. Aristotle throughout emphasises the fact that axioms are self-evident truths, which it is impossible to demonstrate. If, he says, any one should attempt to prove them, it could only be through ignorance. Aristotle therefore would undoubtedly have agreed in Proclus' strictures on Apollonius for attempting to prove the axioms. Proclus gives (p. 194, 25), as a specimen of these attempted proofs by Apollonius, that of the first of the Common Notions. " Let A be equal to B, and the latter to C; I say that A is also equal to C. For, since A is equal to JB, it A B occupies the same space with it ; and since B is equal to C, it occupies the same space with it. Therefore A also occupies the same space with C." Proclus rightly remarks (p. 194, 23) that "the middle term is no more I. C.N 1—3] NOTES ON COMMON NOTIONS 1—3 *»3 intelligible {better known, yvmpifuar€pov} than the conclusion, if it is not actually more disputable." Again (p. 195, 6), the proof assumes two things, (1) that things which "occupy the same space" <tot<k) are equal to one another, and (2) that things which occupy the same space with one and the same thing occupy the same space with one another ; which is to explain the obvious by something much more obscure, for space is an entity more unknown to us than the things which exist in space. Aristotle would also have objected to the proof that it is partial and not general {xatfoXav), since it refers only to things which can be supposed to occupy a space (or take up room), whereas the axiom is, as Froclus says (p. 196, 1), true of numbers, speeds, and periods of time as well, though of course each science uses axioms in relation to its own subject-matter only. Common Notions 2, 3. 2, Kttt lav iiTiia «ra nrpo<rr*9j7, ™ oka JVfw urn. 3. Kai iav otto four lira dtfxHpffljj, ta narnAtHTO/itni fUTtk Ifftt. 7, JftquaU be added to equals, the wholes are equal. 3. //equals he subtracted from equals, the remainders are equal. These two Common Notions are recognised by Heron and Proclus as genuine. The latter is the axiom which is so favourite an illustration, with Aristotle. Following them in the Mss. and editions there came four others of the same type as 1 — 3. Three of these are given by Heiberg in brackets; the fourth he omits altogether. The three are : («) If equals be ad/led to unequals, the wholes are unequal, (b) Things which are double a/ the same thing are equal to oHe another. {c) Things which are halves of the same thing are equal to one another. The fourth, which was placed between (a) and {&), was : (d) If equals be subtracted front unequals, the remainders are unequal. Proclus, in observing that axioms ought not to be multiplied, indicates that all should be rejected which follow from the five admitted by him and appearing in the text above {p. 155). He mentions the second of those just quoted (/') as one of those to be excluded, since it follows from Common Notion 1. Proclus does not mention (a), (c) or (rf); an-Naiml gives (a), (d), {b) and (f), in that order, as Euclid's, adding a note of Simplicius that " three axioms (sentenriae acceptae) only are extant in the ancient manuscripts, but the number was increased in the more recent." (a) stands self-condemned because " unequal " tells us nothing. It is easy to see what is wanted if we refer to 1. 17, where the same angle is added to a greater and a less, and it is inferred that the first sum is greater than the second. So far however as the wording of (a) is concerned, the addition of equal to greater and less might be supposed to produce less and greater respectively. If therefore such an axiom were given at all, it should be divided into two. Heiberg conjectures that this axiom may have been taken from the commentary of Pappus, who had the axiom about equals added to unequais quoted below (e); if so, it can only be an unskilful adaptation of some remark of Pappus, for his axiom (e) has some point, whereas (a) is useless. As regards (b), I agree with Tannery in seeing no sufficient reason why, if «4 BOOK I [i. C. N. a— 4 we reject it (as we certainly must), the words in I. 47 " But things which are double of equals are equal to one another " should be condemned as an interpolation. If they were interpolated, we should have expected to find the same interpolation in ). 42, where the axiom is tacitly assumed. I think it quite possible that Euclid may have inserted such words in one case and left them out in another, without necessarily implying either that he was quoting a formal Common Notion of his own or that he had wt included among his Common Notions the particular fact stated as obvious. The corresponding axiom (c) about the halves of equals can hardly be genuine if {b) is not, and Proclus does not mention it. Tannery acutely observes however that, when Heiberg, in 1. 37, 38, brackets words stating that "the halves of equal things are equal to one another" on the ground that axiom (e) was interpolated (although before Theon's time), and explains that Euclid used Common Notion 3 in making his inference, he is clearly mistaken. For, while axiom (b) is an obvious inference from Common Notion 2, axiom (c) is not an inference from Common Notion 3. Tannery says, in a note, that (c) would have to be established by rtdudio ad absurdum with the help of axiom (6), that is to say, of Common Notion 2. But, as the hypothesis in the reductw ad absurdum would be that one of the halves is greater than the other, and it would therefore be necessary to prove that the one whole is greater than the other, while axiom (b) or Common Notion 2 only refers to equals, a little argument would be necessary in addition to the reference to Common Notion 2. I think Euclid would not have gone through this process in order to prove (c), but would have assumed it as equally obvious with (b). Proclus (pp. 197, 6 — 198, 5) definitely rejects two other axioms of the above kind given by Pappus, observing that, as they follow from the genuine axioms, they are rightly omitted in most copies, although Pappus said that they were " on record " with the others (<rvvara.ypiufn<T$at) : (e) If unequa/s be added to equals, the difference between the wholes is equal to the difference between the added parts ; and (/) //equals be added to unequals, the difference between the wholes is equal to the difference bttween (he original unequals. Proclus and Simplicius (in an-Nairizi) give proofs of both. The proof of the former, as given by Simplicius, is as follows : Let AB, CD be equal magnitudes ; and let EB, FD be £ added to them respectively, EB being greater than FD. q I say that AE exceeds CF by the same difference as that by which BE exceeds DF. Cut off from BE the magnitude BG equal to DF. Then, since AE exceeds AG by GE, and AG is equal to CF and BG to DF, AE exceeds CF by the same difference as that by which BE exceeds DF. Common Notion 4. Kcu to. t<f>upyjj(,uvra tir oAAipXa ura aAAr/Xoi? cotcV. Things which coincide with one another are equal to one anotJur. The word itjmp/Aiiltiv, as a geometrical term, has a different meaning according as it is used in the active or in the passive. In the passive, <£af>/u>£«rltu, it means "to be applied to" without any implication that the applied figure will exactly fit, or coincide with, the figure to which it is applied ; on the other hand the active 'v^ap^t^w is used intransitively and means " to i. C. N. 4] NOTES ON COMMON NOTIONS 2—4 225 lit exactly," " to coincide with." In Euclid and Archimedes t<#>a/)^o{<«' is constructed with «ri and the accusative, in Pappus with the dative. On Common Notion 4 Tannery observes that it is incontestably geometrical in character, and should therefore have been excluded from the Common Notions; again, it is difficult to see why it is not accompanied by its converse, at all events for straight lines (and, it might be added, angles also), which Euclid makes use of in I. 4. As it is, says Tannery, we have here a definition of geometrical equality more or less sufficient, but not a real axiom. It is true that Froclus seems to recognise this Common Notion and the next as proper axioms in the passage (p. 196, 15 — 21) where he says that we should not cut down the axioms to the minimum, as Heron does in giving only three axioms; but the statement seems to rest, not upon authority, but upon an assumption that Euclid would state explicitly at the beginning all axioms subsequently used and not reducible to others unquestionably included. Now in 1. 4 this Common Notion is not quoted ; it is simply inferred that " the base BC will coincide with EF, and will be equal to it." The position is therefore the same as it is in regard to the statement in the same proposition that, "if... the base BC does not coincide with EF, two straight lines will enclose a spate : which is impossible " ; and, if we do not admit that Euclid had the axiom that " two straight lines cannot enclose a space," neither need we infer that he had Common Notion 4. I am therefore inclined to think that the latter is more likely than not to be an interpolation. It seems clear that the Common Notion, as here formulated, is intended to assert that superposition is a legitimate way of proving the equality of two figures which have the necessary parts respectively equal, or, in other words, to serve as an axiom of congruence. The phraseology of the propositions, e.g. 1. 4 and 1. 3, in which Euclid employs the method indicated, leaves no room for doubt that he regarded one figure as actually moved and placed upon the other. Thus in 1. 4 he says, " The triangle ABC being applied (t<£apfu>{o(ia>ov) to the triangle DEF, and the point A being placed {rJltjitvov) upon the point D, and the straight line AB on DE, the point B will also coincide with E because AB is equal to DE"; and in 1. 8, "If the sides BA, AC do not coincide with ED, DF, but fall beside them (take a different position, TapaAAafowrtv), then " etc. At the same time, it is clear that Euclid disliked the method and avoided it wherever he could, e.g. in 1. 26, where he proves the equality of two triangles which have two angles respectively equal to two angles and one side of the one equal to the corresponding side of the other. It looks as though he found the method handed down by tradition (we can hardly suppose that, if Thales proved that the diameter of a circle divides it into two equal parts, he would do so by any other method than that of superposition), and followed it, in -the few cases where he does so, only because he had not been able to see his way to a satisfactory substitute. But seeing how much of the Elements depends on 1. 4, directly or indirectly, the method can hardly be regarded as being, in Euclid, of only subordinate importance , on the contrary, it is fundamental. Nor, as a matter of fact, do we find in the ancient geometers any expression of doubt as to the legitimacy of the method. Archimedes uses it to prove that any spheroidal figure cut by a plane through the centre is divided into two equal parts in respect of both its surface and its volume; he also postulates in Equilibrium of Planes 1. that "when equal and similar plane figures coincide if applied to one another, their centres of gravity coincide also." Killing {Einfuhrung in die Grundlagen der Geometric, 11. pp. 4, 5) 2 26 BOOK I [i. C. M 4 contrasts the attitude of the Greek geometers with that of the philosophers, who, he says, appear to have agreed in banishing motion from geometry altogether. In support of this he refers to the view frequently expressed by Aristotle that mathematics has to do with immovable objects {wivr/rd), and that only where astronomy is admitted as part of mathematical science is motion mentioned as a subject for mathematics. Cf. Mttaph. 989 b 32 "For mathe- matical objects are among things which exist apart from motion, except such as relate to astronomy"; Metapk. rofi4 a 30 "Physics deals with things which have in themselves the principle of motion ; mathematics is a theoretical science and one concerned with things which are stationary (^mfovto) but not separable" (sc. from matter); in Pkysies if. z, 193 b 34 he speaks of the subjects of mathematics as "in thought separable from motion." But I doubt whether in Aristotle's use of the words " immovable," " with- out motion " etc. as applied to the subjects of mathematics there is any implication such as Killing supposes. We arrive at mathematical concepts by abstraction from material objects; and just as we, in thought, eliminate the matter, so according to Aristotle we eliminate the attributes of matter as such, e.g. qualitative change and motion. It does not appear to me that the use of " immovable " in the passages referred to means more than this. I do not think that Aristotle would have regarded it as illegitimate to move a geometrical figure from one position to another ; and I infer this from a passage in De caelo lit. 1 where he is criticising "those who make up every body that has an origin by putting together plants, and resolve it again into planes." The reference must be to the Timaeus (54 b sqq.) where Plato evolves the four elements in this way. He begins with a right-angled triangle in which the hypotenuse is double of the smaller side; six of these put together in the proper way produce one equilateral triangle. Making solid angles with {a) three, (6) four, and (c) five of these equilateral triangles respectively, and taking the requisite number of these solid angles, namely four of (a), six of (&) and twelve of (c) respectively, and putting them together so as to form regular solids, he obtains (a) a tetrahedron, (/J) an octahedron, (v) an icosahedron respectively. For the fourth element (earth), four isosceles right-angled triangles are first put together so as to form a square, and then six of these squares are put together to form a cube. Now, says Aristotle (299 b 23), "it is absurd that planes should only admit of being put together so as to touch in a line; for just as a line and a line are put together in both ways, lengthwise and breadthwise, so must a plane and a plane. A line can be combined with a line in the sense of being a line superposed, and not added"; the inference being that a plane can be superposed on &plane. Now this is precisely the sort of motion in question here; and Aristotle, so far from denying its permissibility, seems to blame Plato for not using it. Cf. also Physits v. 4, 228 b 25, where Aristotle speaks of "the spiral or other magnitude in which any part will not coincide with any other part," an where superposition is obviously contemplated. Motion without deformation. It is well known that Helmholtz maintained that geometry requires us to assume the actual existence of rigid bodies and their free mobility in space, whence he inferred that geometry is dependent on mechanics. Veronese exposed the fallacy in this {Fondamenti ii geometria, pp. xxxv — xxxvi, 239 — 240 note, 615 — 7), his argument being as follows. Since geometry is concerned with empty space, which is immovable, it would be at least strange if it was necessary to have recourse to the real motion of bodies for a definition, ]. C. N. 4] NOTE ON COMMON NOTION 4 xa-j and for the proof of the properties, of immovable space. We must distinguish the intuitive principle of motion in itself from that of motion without deforma- tion. Every point of a figure which moves is transferred to another point in space. " Without deformation " means that the mutual relations between the points of the figure do not change, but the relations between them and other figures do change (for if they did not, the figure could not move). Now consider what we mean by saying that, when the figure A has moved from the position A, to the position A it the relations between the points of A in the position A, are unaltered from what they were in the position A u are the same in fact as if A had not moved but remained at A t . We can only say that, judging of the figure (or the body with its physical qualities eliminated) by the impressions it produces in us during its movement, the impressions produced in us in the two different positions (which are in time distinct) are equal. In fact, we are making use of the notion of equality between two distinct figures. Thus, if we say that two bodies are equal when they can be superposed by means of movement without deformation, we are com- mitting a petitio principii. The notion of the equality of spaces is really prior to that of rigid bodies or of motion without deformation. HelmholU supported his view by reference to the process of measurement in which the measure must be, at least approximately, a rigid body, but the existence of a rigid body as a standard to measure by, and the question how we discover two equal spaces to be equal, are matters of no concern to the geometer. The method of superposition, depending on motion without deformation, is only of use as a practical test ; it has nothing to do with the theory of geometry. Compare an acute observation of Schopenhauer {Die Welt als (Vi/le, 2 ed- 1844, 11. p. 130) which was a criticism in advance of Helmholtz' theory : "I am surprised that, instead of the eleventh axiom [the Parallel-Postulate], the eighth is not rather attacked : ' Figures which coincide (sich decken) are equal to one another.' For coincidence (das Sichdecken) is either mere tautology, or something entirely empirical, which belongs, not to pure intuition (Anschauung), but to external sensuous experience. It presupposes in fact the mobility of figures ; but that which is movable in space is matter and nothing else. Thus this appeal to coincidence means leaving pure space, the sole element of geometry, in order to pass over to the material and empirical." Mr Bertrand Russell observes {Encyclopaedia Britannica, Suppl. Vol. 4, 1Q02, Art. " Geometry, non-Euclidean ") that the apparent use of motion here is deceptive ; what in geometry is called a motion is merely the transference of our attention from one figure to another. Actual superposition, which is nominally employed by Euclid, is not required; all that is required is the transference of our attention from the original figure to a new one defined by the position of some of its elements and by certain properties which it shares with the original figure. If the method of superposition is given up as a means of defining theoreti- cally the equality of two figures, some other definition of equality is necessary. But such a definition can be evolved out of empirical or practical observation of the result of superposing two material representations of figures. This is done by Veronese {Elementi di geometria, 1904) and Ingrami {Element i di geometria, 1904). Ingrami says, namely (p. 66); " If a sheet of paper be folded double, and a triangle be drawn Upon it and then cut out, we obtain two triangles superposed which we in practice call equal. If points A, B, C, D ... be marked on one of the triangles, then, when we place this triangle upon the other (so as to coincide with it), we see 1*8 BOOK I [i. C. If. 4 chat each of the particular points taken on the first is superposed on one particular point of the second in such a way that the segments AB, AC, AD, BC, BD, CD, ... ace respectively superposed on as many segments in the second triangle and are therefore equal to them respectively. In this way we justify the following "Definition of equality. " Any two figures whatever will be called equal when to the points of one the points of the other can be made to correspond univocally [i.e. every one point in one to one distinct point in the other and vice versa] in such a way that the segments which join the points, two and two, in one figure are respectively equal to the segments which join, two and two, the corresponding points in the other." Ingram! has of course previously postulated as known the signification of the phrase equal (reciiUneat) segments, of which we get a practical notion when we can place one upon the other or can place a third movable segment successively on both. New systems of Congruence-Postulates. In the fourth Article of Questioni riguardanti ie matematiehe etementari, I., pp. 93 — 122, a review is given of three different systems : (i) that of Pasch in Vorlesungen titer neuere Geometrie, 1882, p. 101 sqq., (3) that of Veronese according to the Fondamenti di geometria, 1891, and the Ekmcnti taken together, (3) that of Hilbert (see Grundlagen der Geometric, 1903, pp. 7—15). These systems differ in the particular conceptions taken by the three authors as primary, (t) Pasch considers as primary the notion of congruence or equality between any figures which are made up of a finite number 0/ points only. The definitions of congruent segments and of congruent angles have to be deduced in the way shown on pp. 102 — 103 of the Article referred to, after which Eucl. 1. 4 follows immediately, and Eucl. 1. 26 (1) and 1. 8 by a method recalling that in Eucl. 1. 7, 8. (2) Veronese takes as primary the conception of congruence between segments (rectilineal). The transition to congruent angles, and thence to triangles is made by means of the following postulate: "Let AB, ^Cand AB, A'C be two pairs of straight lines intersecting at A, A', and let there be determined upon them the congruent segments AB, A'ff and the congruent segments AC, A'C ; then, if BC, BC are congTuent, the two pairs of straight lines are con- gruent." <i) Hilbert takes as primary the notions of congruence between both segments and angles. It is observed in the Article referred to that, from the theoretical stand- point, Veronese's system is an advance upon that of Pasch, since the idea of congruence between segments is more simple than that of congruence between any figures; but, didactically, the development of the theory is more compli- cated when we start from Veronese's system than when we start from that of Pasch. The system of Hilbert offers advantages over both the others from the point of view of the teaching of geometry, and I shall therefore give a short account of his system only, following the Artiole above quoted. i. C. JV. 4] NOTE ON COMMON NOTION 4 339 Hilbcrt's system. The following are substantially the Postulates laid down, (1) If one segment is congruent toith another, the second is also congruent with the first. (2) If an angle is congruent with another angle, the second angle is also congruent with the first. (3) Two segments congruent with a third are congruent with one another. (4) Two angles congruent with a third are congruent with one another. (5) Any segment AB is congruent with itself, independently of its sense. This we may express symbolically thus : AB=AB= BA. (6) Any angle (ab) is congruent with itself, independently of its sense. This we may express symbolically thus : (ab) = (ab) = (6a). (7) On any straight line x\ starting from any one of its points A', and on each side of it respectively, there exists one and only one segment congruent with a segment AB belonging to the straight tine 1. (&) Given a ray a, issuing from a paint O, in any plane which contains it and on each of the two sides of it, there exists one and only one ray b issuing from O such that the angle (ab) is congruent with a given angle (a'b'). (9) If AB, BC are two consecutive segments of the same straight line r (segments, that is, having an extremity and no other point common), and A'B', B'C two consecutive segments on another straight line r", and if AB E A'B', BC 3 B'C, then AC=A'C. {10) If (ah), (be) are two consecutive angles in the same planer (angles, that is, having the vertex and one side common), and (a'b'), (b'c') two consecu- tive angles in another plane ■*■', and if (ab) £ (a'b'), (be) = (b'c'), then (ac) = (a'c'). (n) If two triangles have two sides and the included angles respectively congruent, they have also their third sides congruent as well as the angles opposite to the congruent sides respectively. As a matter of fact, Hilbert's postulate corresponding to (n) does not assert the equality of the third sides in each, but only the equality of the two remaining angles in one triangle to the two remaining angles in the other respectively. He proves the equality of the third sides (thereby completing the theorem of Eucl. L 4) by reductio ad absurdum thus. Let ABC, A'B'C A A' be the two triangles which have the S\ jr\ sides AB, AC respectively congruent ^r \ jr jl with the sides A B', A'C and the j^ \ y^ ; \ included angle at A congruent with ^— £ gf- £-jy the included angle at A'. Then, by Hilbert's own postulate, the angles ABC, A'B'C are congruent, as also the angles ACB, A'C'ff. If BC is not congruent with B'C, let D be taken on BC such that BC, BD are congruent and join A'D. 330 BOOK I [i. C. N. 4 Then the two triangles ABC, A'ffD have two sides and the included angles congruent respectively ; therefore, by the same postulate, the angles BAC, SA'D are congruent But the angles BAC, B'A'C' are congruent; therefore, by (4) above, the angles B'A'C 1 , BAD are congruent : which is impossible, since it contradicts (8) above. Hence BC, B'C cannot but be congruent. Eucl. 1. 4 is thus proved ; but it seems to be as well to include all of that theorem in the postulate, as is done in (11) above, since the two parts of it are equally suggested by empirical observation of the result of one superposition. A proof similar to that just given immediately establishes Eucl. 1. 26 (1), and Hilbert next proves that If two angles ABC, A' B'C' are congruent with one another, their supple- mentary angles CBD, C'B'D' are also congruent with one another. We choose A, D on one of the straight lines forming the first angle, and A', & on one of those forming the second angle, and again C, C on the other * Vi 0' straight lines forming the angles, so that A'B is congruent with AB, C'ff with CB, and Uff with DB. The triangles ABC, A' B'C are congruent, by (11) above; and AC is congruent with A'C, and the angle CAB with the angle C'A'B 1 . Thus, AD, A'D' being congruent, by (9), the triangles CAD, C'A'D are also congruent, by ( 1 1 ) ; whence CD is congruent with CD?, and the angle ADC with the angle ADC. Lastly, hy (n), the triangles CDS, CDS are congruent, and the angles CBD, C'B'D are thus congruent. Hilbert's next proposition is that Given that the angle (h, k) in the plane a is congruent ■with the angle (h', k') in the plane a', and that 1 is a half-ray in the plane a. starting from the vertex of the angle (h, k) and lying within that angle, there always exists a half-ray V in the second plane a', starting from the vertex of the angle {h', k') and lying within that angle, such that (h,l)s(h',l'), and(k, l) = (k',I'). If O, C are the vertices, we choose points A, B on h, k, and points A, ff on H, k' respectively, such that OA, OA are congruent and also OB, Off. The triangles OAB, OAff are then congruent ; and, if / meets AB in C, : can determine C on A'B such that AC is congruent with AC. Then f drawn from through C is the half-ray required. I. C. N. 4] NOTE ON COMMON NOTION 4 S31 The congruence of the angles (h, I), (A', /) follows from (11) directly, and that of {k, I) and {X, t) follows in the same way after we have inferred by means of (9) that, AB, AC being respectively congruent with A'B 1 , A'C, the difference BC is congruent with the difference B C'. It is by means of the two propositions just given that Hilbert proves that All right angles are congruent with one another. Let the angle BAD be congruent with its adjacent angle CAD, and likewise the angle BA'D congruent with its adjacent angle CA'D All four angles are then right angles. A' C If the angle B'A'D is not congruent with the angle BAD, let the angle with AB for one side and congruent with the angle B'A'D be the angle BAD', so that AD' falls either within the angle BAD or within the angle DAC. Suppose the former. By the last proposition but one (about adjacent angles), the angles BAD, BAD' being congruent, the angles CA'D, CAD' are congruent. Hence, by the hypothesis and postulate (4) above, the angles BAD' , CAD" are alio congruent. And, since the angles BAD, CAD are congruent, we can find within the ingle CAD a half-ray CAD"' such that the angles BAD', CAD" are congruent, and likewise the angles DAD', DAD" (by the last proposition). But the angles BAD', CAD" were congruent (see above); and it follows, by (4), that the angles CAD', CAD" are congruent : which is impossible, since it contradicts postulate (8), Therefore etc. Euclid 1. s follows directly by applying the postulate (n) above to ABC, ACB as distinct triangles. Postulates (9), (10) above give in substance the proposition that "the sums or differences of segments, or of angles, respectively equal, are equal." Lastly, Hilbert proves Eucl. 1. 8 by means of the theorem of Eucl. 1. 5 and the proposition just stated as applied to angles. ABC, A'BC being the given triangles with three sides respectively congruent, we suppose an angle CBA" to be deter- mined, on the side of BC opposite to A, congruent with the angle A'BC, and we make BA" equal to A'B. The proof is obvious, being equivalent to the alternative proof often given in our text-books for Eucl. I. 8. a 3 « BOOK I [[. C. N. 5 Common Notion 5. koI to oAov tou ttipavs fLttfcav [f tTTtv], J>fe icAoA w greater than the part. Prod us includes this " axiom " on the same ground as the preceding one. I think however there is force in the objection which Tannery takes to it, namely that it replaces a different expression in Eucl. f. 6, where it is stated that " the triangle DBC will be equal to the triangle A CB, the /ess to the greater: which is absurd." The axiom appears to be an abstraction or generalisation substituted for an immediate inference from a geometrical figure, but it takes the form of a sort of definition of whole and part. The probabilities seem to be against its being genuine, notwithstanding Proclus' approval of it Clavius added the axiom that the whole is the equal to the sum of its parts. Other Axioms introduced after Euclid's time. [9] Two straight lines do not enclose (or can tain) a space. Proclus {p. 196, 21) mentions this in illustration of the undue multiplication of axioms, and he points out, as an objection to it, that it belongs to the subject matter of geometry, whereas axioms are of a general character, and not peculiar to any one science. The real objection to the axiom is that it is unnecessary, since i.he fact which it states is included in the meaning of Postulate i. It was nr> doubt taken from the passage in 1. 4, "if. the base BC does not coincide with the base EF, two straight tines wilt enclose a space : which is impossible"; and we must certainly regard it as an interpolation, notwithstanding that two of the best mss. have it after Postulate 5, and one gives it as Common Notion 9. Pappus added some others which Proclus objects to (p. 198, 5) because they are either anticipated in the definitions or follow from them. (g) All the parts of a plane, or of a straight line, coincide with one another. \K) A point divides a line, a line a surface, and a surface a solid; on which Proclus remarks that everything is divided by the same things as those by which it is bounded. An-Nairizi {ed. Besthorn-Heiberg, p. 31, ed. Curtze, p. 38) in his version of this axiom, which be also attributes to Pappus, omits the reference to solids, but mentions planes as a particular case of surfaces. " (a) A Surface cuts a surface in a line ; fj8) If two surfaces which cut one another are plane, they cut one another in a straight line ; (y) A line cuts a line in a point (this last we need in the first proposition}." (A) Magnitudes are susceptible of the infinite {or unlimited) both by way of addition and by way of successive diminution, but in both cases potentially only (to mrttpov fv rots fnyi$ttrw iirriv got Tp xpoaGitrtt Kal rp iiittai0atpio , u, oW ifLtt Si imLrtpav). An-Nairizi's version of this refers to straight lines and plane surfaces only : "as regards the straight line and the plane surface, in consequence of their evenness, it is possible to produce them indefinitely. This "axiom" of Pappus, as quoted by Proclus, seems to be taken directly from the discussion of to avtipov in Aristotle, Physics ill. 5 — 8, even to the wording, for, while Aristotle uses the term division {huaiptatsi) most frequently as the antithesis of addition (avrBttrn), he occasionally speaks of subtraction (AAJHupttrn) and diminution (icaflaipHrii). Hankel (Zur Geschichte der Mathe- matik im Alterthum und Mittelalter, 1874, pp. 119 — 120) gave an admirable 1. Axx.] ADDITIONAL AXIOMS 233 summary of Aristotle's views on this subject ; and they are stated in greater detail in Gorland, AristoteUs und die Mathematik, Marburg, 1899, pp. 157— 183. The infinite or unlimited (airttpov) only exists potentially {Sura/i«), not in actuality (Ivtpytiq.). The infinite is so in virtue of its endlessly changing into something else, like day or the Olympic Games {Phys. m. 6, 206 a 15 — 2 S)- The infinite is manifested in different forms in time, in Man, and in the division of magnitudes. For, in general, the infinite consists in something new being continually taken, that something being itself always finite but always different. Therefore the infinite must not be regarded as a particular thing (toS« t»), as man, house, but as being always in course of becoming or decay, and, though finite at any moment, always different from moment to moment. But there is the distinction between the forms above referred to that, whereas in the case of magnitudes what is once taken remains, in the case of time and Man it passes or is destroyed but the succession is unbroken. The case of addition is in a sense the same as that of division ; in the finite magnitude the former takes place in the converse way to the latter ; for, as we see the finite magnitude divided ad infinitum, so we shall find that addition gives a sum tending to a definite limit. I mean that, in the case of a finite magnitude, you may take a definite fraction of it and add to it (continually) in the same ratio ; if now the successive added terms do not include one and the same magnitude whatever it is [i.e. if the successive terms diminish in geometrical progression], you will not come to the end of the finite magnitude, but, if the ratio is increased so that each term does include one and the same magnitude whatever it is, you will come to the end of the finite magnitude, for every finite magnitude is exhausted by continually taking from it any definite fraction whatever. Thus in no other sense does the infinite exist, but only in the sense just mentioned, that is, potentially and by way of diminution (106 a 25 — b 13). And in this sense you may have potentially infinite addition, the process being, as we say, in a manner, the same as with division ad infinitum : for in the case of addition you will always be able to find some* thing outside the total for the time being, but the total will never exceed every definite (or assigned) magnitude in the way that, in the direction of division, the result will pass every definite magnitude, that is, by becoming smaller than it. The infinite therefore cannot exist even potentially in the sense of exceeding every finite magnitude as the result of successive addition (206 b 16 — 22). It follows that the correct view of the infinite is the opposite of that commonly held : it is not that which has nothing outside it, but that which always has something outside it (206 b 33 — 207 a r). Contrasting the case of number and magnitude, Aristotle points out that (1) in number there is a limit in the direction of smallness, namely unity, but none in the other direction : a number may exceed any assigned number however great ; but (2) with magnitude the contrary is the case : you can find a magnitude smaller than any assigned magnitude, but in the other direction there is no such thing as an infinite magnitude (207 b r — 5). The tatter assertion he justified by the following argument. However large a thing can be potentially, it can be as large actually. But there is no magnitude perceptible to sense that is infinite. Therefore excess over every assigned magnitude is an impossibility j otherwise there would be something larger than the universe (oipavot) {207 b 17—21). Aristotle is aware that it is essentially of physical magnitudes that he is speaking. He had observed in an earlier passage (PAys, in, 5, 204 a 34) that it ts perhaps a more general inquiry that would be necessary to determine *34 BOOK ! [i. Axx. whether the infinite is possible in mathematics, and in the domain of thought and of things which have no magnitude ; but he excuses himself from entering upon this inquiry on the ground that his subject is physics and sensible objects. He returns however to the bearing of his conclusions on mathematics in m. 7, 207 b 2j : "my argument does not even rob mathematicians of their study, although it denies the existence of the infinite in the sense of actual existence as something increased to such an extent that it cannot be gone through (dSwfiTifrov) ; for, as it is, they do not even need the infinite or use it, but only require that the finite (straight line) shall be as long as they please; and another magnitude of any size whatever can be cut in the same ratio as the greatest magnitude. Hence it will make no difference to them for the purpose of demonstration." I^astly, if it should be urged that the infinite exists in thought, Aristotle replies that this does not involve its existence in fact. A thing is not greater than a certain size because it is conceived to be so, but because it is; and magnitude is not infinite in virtue of increase in thought (ao8 a 16 — zz). Hankel and Gorland do not quote the passage about an infinite series of magnitudes {206 b 3—13) included in the above paraphrase; but I have thought that mathematicians would be interested in the distinct expression of Aristotle's view that the existence of an infinite series the terms of which are magnitudes is impossible unless it is convergent, and (with reference to Riemann's developments) in the statement that it does not matter to geometry if the straight line is not infinite in length, provided that it is as long as we please. Aristotle's denial of even the potential existence of a sum of magnitudes which shall exceed every definite magnitude was, as he himself implies, in conflict with the lemma or assumption used by Eudoxus (as we infer from Archimedes) to prove the theorem about the volume of a pyramid. The lemma is thus stated by Archimedes (Quadrature of a parabola, preface): " The excess by which the greater of two unequal areas exceeds the less can, if it be continually added to itself, be made to exceed any assigned finite area." We can therefore well understand why, a century later, Archimedes felt it necessary to justify his own use of the lemma as he does in the same preface ; " The earlier geometers too have used this lemma : for it is by its help that they have proved that circles have to one another the duplicate ratio of their diameters, that spheres have to one another the triplicate ratio of their diameters, and so on. And, in the result, each of the said theorems has been accepted no less than those proved without the aid of this lemma." Principle of continuity. The use of actual construction as a method of proving the existence ot figures having certain properties is one of the characteristics of the Elements. Now constructions are effected by means of straight lines and circles drawn in accordance with Postulates 1 — 3 ; the essence of them is that such straight lines and circles determine by their intersections other points in addition to those given, and these points again are used to determine new lines, and so on. This being so, the existence of such points of intersection must be postulated or proved in the same way as that of the lines which determine them. Yet there is no postulate of thfs character expressed in Euclid except Post J. This postulate asserts that two straight lines meet if they satisfy a certain condition. The condition is of the nature of a Stop«r>«5s (discrimination, or condition of possibility) in a problem ; and, if the existence of the point of i. Axx.l PRINCIPLE OF CONTINUITY 335 intersection were not granted, the solutions of' problems in which the points of intersection of straight lines are used would not in general furnish the required proofs of the existence of the figures to be constructed. But, equally with the intersections of straight lines, the intersections of circle with straight line, and of circle with circle, are used in constructions. Hence, in addition to Postulate 5, we require postulates asserting the actual existence of points of intersection of circle with straight line and of circle with circle. In the very first proposition the vertex of the required equilateral triangle is determined as one of the intersections of two circles, and we need therefore to be assured that the circles will intersect. Euclid seems to assume it as obvious, although it is not so; and he makes a similar assumption in 1. 2 2. It is true that in the latter case Euclid adds to the enunciation that two of the given straight lines must be together greater than the third ; but there is nothing to show that, if this condition is satisfied, the construction is always possible. In 1. 12, in order to be sure that the circle with a given centre will intersect a given straight line, Euclid makes the circle pass through a point on the side of the line opposite to that where the centre is. It appears therefore as if, in this case, he based his inference in some way upon the definition of a circle combined with the fact that the point within it called the centre is on one side of the straight line and one point of the circumference on the other, and, in the case of two intersecting circles, upon similar con- siderations. But not even in Book hi., where there are several propositions about the relative positions of two circles, do we find any discussion of the conditions under which two circles have two, one, or no point common. The deficiency can only be made good by the Principle of Continuity. Killing {Einfiihrung in die Grundlagen der Geometric, 11. p 43) gives the following forms as sufficient for most purposes (a) Suppose a line belongs entirely to a figure which is divided into two parts; then, if the line has at least one point common with each part, it must also meet the boundary between the parts; or {b) If a point moves in a figure which is divided into two parts, and if it belongs at the beginning of the motion to one part and at the end of the motion to the other part, it must during the motion arrive at the boundary between the two parts. I n the Questioni riguardanti le matematuhe elemeniari, l.,Art.s,pp. 123—143, the principle of continuity is discussed with special reference to the Postulate of Dedektnd, and it is shown, first, how the Postulate may be led up to and, secondly, how it may be applied for the purposes of elementary geometry. Suppose that in a segment A B of a straight line a point C determines two segments AC, CB. If we consider the point Cas belonging to only one of the two segments A C, CB, we have a division of the segment AB into two parts with the following properties. 1. Every point of the segment AB belongs to one of the two parts. 2. The point A belongs to one of the two parts (which we will call the first) and the point B to the other; the point C may belong indifferently to one or the other of the two parts according as we choose to premise. 3. Every point of the first part precedes every point of the second in the order AB of the segment. (For generality we may also suppose the case in which the point C falls at A or at B. Considering C, in these cases respectively, as belonging to the first or second part, we still have a division into parts which have the properties above enunciated, one part being then a single point A or B.) J36 BOOK 1 [i. Axx. Now, considering carefully the inverse of the above proposition, we see that it agrees with the idea which we have of the continuity of the straight line. Consequently we are induced to admit as a postulate the following. If a segment of a straight lint AB is divided into two parts so that (i ) every point of the segment AB belongs to one of the parts, (2) the extremity A belongs to the first part and B to the second, and (3) any point whatever of the first part precedes any point whatever of the second part, in the order AB of the segment, there exists a point C of the segment AB {which may belong either to one part or to the other) such that every point of AB that precedes C belongs to the first part, and every point of AB that follows C belongs to the second part in the division originally assumed. (If one of the two parts consists of the single point A or B, the point C is the said extremity A or B of the segment.) This is the Postulate of Dedekind, which was enunciated by Dedekind himself in the following slightly different form (Stetigkcit unci irrationale Zahlen, 187s, new edition 1905, p. 11). " If all points of a straight line fall into hvo classes such that every point of the first class lies to the left of every point of the second class, there exists one and only one point which produces this division of all the points into two daises, this division of the straight line into two parts." The above enunciation may be said to correspond to the intuitive notion which we have that, if in a segment of a straight line two points start from the ends and describe the segment in opposite senses, they meet in a point. The point of meeting might be regarded as belonging to both parts, but for the present purpose we must regard it as belonging to one only and subtracted from the other part. Application of Dedckind's postulate to angles. If we consider an angle less than two right angles bounded by two rays a, b, and draw the straight line connecting A, a point on a, with B, a point on b, we see that all points on the finite segment AB correspond univocally to all the rays of the angle, the point corresponding to any ray being the point in which the ray cuts the segment AB ; and if a ray be supposed to move about the vertex of the angle from the position a to the position b, the corresponding points of the segment AB are seen to follow in the same order as the corresponding rays of the angle (ab). Consequently, if the angle (ab) is divided into two parts so that (1) each ray of the angle (ab) belongs to one of the two parts, (t) the outside ray a belongs to the first part and the ray b to the second, (3) any ray whatever of the first part precedes any ray whatever of the second part, the corresponding points of the segment AB determine tteo parts of the segments such that (1) every point of the segment AB belongs to one of the two parts, (2) the extremity A belongs to the first part and B to the second, (3) any point whatever of the first part precedes any point whatever of the second. But in that case there exists a point C of AB {which may belong to one or the other of the two parts) such that every point of AB that precedes C belongs to the first part and every point of AB that follows C belongs to the second part. I. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE ^37 Thus exactly the same thing holds of c, the ray corresponding to C, with reference to the division of the angle {ad) into two parts. It is not difficult to extend this to an angle (ai) which is either flat or greater than two right angles; this is done (Vitali, op. cit. pp. 126—127) by supposing the angle to be divided into two, {ad), {di>), each less than two right angles, and considering the three cases in which (1) the ray d is such that all the rays that precede it belong to the first patt and those which follow it to the second part, (2) the ray d is followed by some rays of the first part, (3) the ray d is preceded by some rays of the second part. Application to circular arcs. If we consider an arc AB of a circle with centre O, the points of the arc correspond uni vocally, and in the same order, to the rays from the point O passing through those points respectively, and the same argument by which we passed from a segment of a straight line to an angle can be used to make the transition from an angle to an arc. Intersections of a straight line with a circle. It is possible to use the Postulate of Dedekind to prove that If a straight line has one point inside and one point outside a circle, it has two points common with the circle. For this purpose it is necessary to assume (1) the proposition with reference to the perpendicular and obliques drawn from a given point to a given straight line, namely that of all straight lines drawn from a given point to a given straight line the perpendicular is the shortest, and of the rest (the obliques) that is the longer which has the longer projection upon the straight line, while those are equal the projections of which are equal, so that for any given length of projection there are two equal obliques and two only, one on each side of the perpendicular, and (2) the proposition that any side of a triangle is less than the sum of the other two. Consider the circle {(7) with centre 0, and a straight line {r) with one point A inside and one point B outside the circle. By the definition of the circle, if R is the radius, OA<R, OB>R. Draw OP perpendicular to the straight line r. Then OP< OA, so that OP is always less than R, and P is therefore within the circle C. Now let us fix our attention on the finite segment AB of the straight line r. It can be divided into two parts, (1) that containing all the points H for which OIf< R (i.e. points inside C), and (2) that containing all the points K for which OK £ R (points outside C or on the circumference of C). Thus, remembering that, of two obliques from a given point to a given straight line, that is greater the projection of which is greater, we can assert that all the points of the segment PB which precede a point inside C are inside C, and those which follow a point on the circumference of C or outside C are outside C. Hence, by the Postulate of Dedekind, there exists on the segment PB a 238 BOOK I [1. Axx. point M such that all the points which precede it belong to the first part and those which follow it to the second part. I say that M is common to the straight line r and the circle C, or OM=R. For suppose, e.g., that OM <.R. There will then exist a segment (or length) a less than the difference between R and OM. Consider the point Af, one of those which follow M, such that MAT is equal to a. Then, because any side of a triangle is less than the sum of the Other two, OM' < OM+ MM'. But OM+ MM' = OM+<r<R, whence OM' < R, which is absurd. A similar absurdity would follow if we suppose that OM > R. Therefore OM mist be equal to R. It is immediately obvious that, corresponding to the point Mor\ the segment PS which is common to r and C, there is another point on r which has the same property, namely that which is symmetrical to M with respect to P. And the proposition is proved. Intersections of two circles. We can likewise use the Postulate of Dedekind to prove that If in a given plane a circle C has one paint X inside and one point Y outside another circle C ', the two circles intersect in two points. We must first prove the following Lemma. If O, 0' are the centres of two circles C, C, and R, R' their radii respectively, the straight line 00' meets the circle C in two points A, S, one of which is inside C" and the other outside it. Now one of these points must fall (r) on the prolongation of 00 beyond O or {2) on 00 itself or {3) on the prolongation of OO' beyond 0. (1) First, suppose A to lie on OO pro- duced. Then A0=AO+ 00' = JP+ OO" (a). But, in the triangle OO Y, 0Y<OY+O0, and, since 0Y>R\ OY=R, R<R+ Off. It follows from (a) that A0>R; and therefore lies outside C". (2) Secondly, suppose A to lie on 00. Then 00 = OA + A0 = R + A0 ...($). From the triangle 00 X we have O0<OX+0X, and, since OX-R, 0X<R, it follows that 00'<R + R, whence, by {£), A0 < R, so that A lies inside C i. Axx.] APPLICATIONS OF DEDEKIND'S POSTULATE 2 39 Consider {3) Thirdly, suppose A to lie on 00 produced. Then R = OA = 00 + O'A (7). And, in the triangle 00 X, ox<oa+0x, w that is R < 00 + OX, B f whence, by (y), 00 + O'A < Off + 0X, or O'A < 0X, and A lies inside C. It is to be observed that one of the two points A, B is in the position of case (1) and the other in the position of either case {2) or case (3) : whence we must conclude that one of the two points A, B is inside and the other outside the circle C. Proof of theorem. The circle C is divided by the points A, B into two semicircles, one of them, and suppose it to be described by a point moving from A to B. Take two separate points P, Q on it and, to fix our ideas, suppose that P precedes Q. Comparing the triangles O0P, OaQ, we observe that one side 00 is common, OP is equal to OQ, and the angle PO0 is less than the angle QO0. Therefore 0P< 0Q. Now, considering the semicircle APQB as divided into two parts, so that the points of the first part are inside the circle C, and those of the second part on the circumference of C or outside it, we have the conditions necessary for the applicability of the Postulate of Dedekind (which is true for arcs of circles as for straight lines) ; whence there exists a point M separating the two parts. I say that 0M=R". For, if not, suppose 0M < R 1 . If then a signifies the difference between R' and 0M, suppose a point if, which follows M, taken on the semicircle such that the chord MM' is not greater than cr (for a way of doing this see below). Then, in the triangle 0MM', 0AT < OM+ MM' < 0M+ <r, and therefore 0M' < R. It follows that M', a point on the arc MB, is inside the circle C": which is absurd. Similarly it may be proved that 0M is not greater than R. Hence 0M=R. [To find a point M' such that the chord MM' is not greater than a, we may proceed thus. Draw from M a straight line MP distinct from OM, and cut off MP on it equal to er/a. a4o BOOK I [i. Axx. Join OP, and draw another radius OQ such that the angle POQ is equal to the angle MOP. q The intersection, M\ of 0@ with the circle satisfies the required condition. For MM' meets OP at right angles in S. Therefore, in the right-angled triangle MSP, MS is not greater than MP (it is less, unless MP coincides with MS, when it is equal). Therefore MS is not greater than <rjt, so that MM' is not greater than <r.\ BOOK I. PROPOSITIONS. Proposition i. On a given finite straight line to construct an equilateral triangle. Let AB be the given finite straight line. Thus it is required to con- s struct an equilateral triangle on the straight line AB. With centre A and distance AB let the circle BCD be described ; [Post. 3] ic again, with centre B and dis- tance BA let the circle ACE be described ; [Post 3] and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. [Post. 1] ts Now, since the point A is the centre of the circle CDB, AC is equal to AB. [Def. 15] Again, since the point B is the centre of the circle CAB, BC is equal to BA. [Def. 15] But CA was also proved equal to AB ; so therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another ; [C. JV, 1] therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are aj equal to one another. 343 BOOK I [l. i Therefore the triangle ABC is equilateral ; and it has been constructed on the given finite straight line AB. (Being) what it was required to do. I. On ■ given finite straight line. The Greek usage differs from ours in that the definite article is employed in such a phrase as this where we have Lhe indefinite, #ri rfli &ot)(taw ttffttat TtrepMnfrw, "on the given finite straight line," i.e. the linite straight line which we choose to take. -3. Let AB be the given finite straight line. To be strictly literal we should have to translate in the reverse order "let the given finite straight line be the (straight line) AB"; but this order is inconvenient in other cases where there is more than one datum, e.g. in the sitting-out of I. 1, "let the given point be A, and the given straight Line BC," the awkward- ness arising from the omission of the verb in the second clause. Hence I have, for clearness' sake, adopted the other order throughout the book. S. let the circle BCD be described. Two things are here to be noted, (1) the elegant and practically universal use of the perfect passive imperative in constructions, ycyp&pt* meaning of course "let it havt ban described" or "suppose it described," ft) the impossi- bility of expressing shortly in a translation the force of the words in their original order. Ktnihet yryai$8vi BTA means literally "let a circle have been described, the (circle, namely, which I denote by) BCD" Similarly we have lower down " let straight lines, (namely) the (straight Lines) CA, CB, be joined," erffftfx&Mrai' eidtiot til PA, PB. There seems to be no practicable alternative, in English, hut to translate as 1 have done in the text. 13. from the point C„.« Euclid is careful to adhere to the phraseology of Postulate I except that he speaks of "joining" {tTi&$x8<#rar) instead of "drawing (yp&fair). He does not allow himself to use the shortened expression " let the straight line FC be joined" (without mention of the points F, C\ until 1. 5. 10. each of the straight lines CA, CB, irmipa tut TA, FB and 14. the three straight lines CA, AB, BC, td rpcU a! 1'A, AB, Br. t have, here and in all simitar expressions, inserted the words "straight Lines " which are not in the Greek. The possession of the inflected definite article enables the Greek to omit the words, but this it not possible in English, and it would scarcely be English to write "each of CA, CB " or "the three CA, AB, BC." It is a commonplace that Euclid has no right to assume, without pre- mising some postulate, that the two circles will meet in a point C. To supply what is wanted we must invoke the Principle of Continuity (see note thereon above, p. 235). It is sufficient for the purpose of this proposition and of 1. 21, where there is a similar tacit assumption, to use the form of postulate suggested by Killing. " If a line [in this case e.g. the circumference A CE] belongs entirely to a figure [in this case a plane] which is divided into two parts [namely the part enclosed within the circumference of the circle BCD and the part outside that circle], and if the line has at least one point common with each part, it must also meet the boundary between the parts [i.e. the circum- ference ACE must meet the circumference BCD]." Zeno's remark that the problem is not solved unless it is taken for granted that two straight lines cannot have a common segment has already been mentioned (note on Post a, p. 100). Thus, if AC, BC meet at F before reaching C, and have the part EC common, the triangle obtained, namely FAB, will not be equilateral, but PA, EB will each be less than AB. But Post 2 has already laid it down that two straight lines cannot have a common segment. Proclus devotes considerable space to this part of Zeno's criticism, but satisfies himself with the bare mention of the other part, to the effect that it. is also necessary to assume that two circumferences (with different centres) cannot have a common part That is, for anything we know, there may be any number of points C common to the two circumferences ACE, BCD. It is not until in. 10 that it is proved that two circles cannot intersect in more ...] PROPOSITION i *43 points than two, so that we are not entitled to assume it here. The most we can say is that it is enough foi the purpose of this proposition if one equilateral triangle can be found with the given base ; that the construction only gives two such triangles has to be left over to be proved subsequently, And indeed we have not long to wait ; for i. 7 clearly shows that on either side of the base AB only one equilateral triangle can be described. Thus 1. 7 gives us the number of solutions of which the present problem is susceptible, and it supplies the same want in 1. 22 where a triangle has to be described with three sides of given length ; that is, 1. 7 furnishes us, in both cases, with one of the essential parts of a complete Stopiff/ios, which includes not only the determination of the conditions of possibility but also the number of solutions {ir&rax&s tyxvp't, Proclus, p. so*, 5). This view of 1. 7 as supplying an equivalent for 111. ro absolutely needed in 1. 1 and 1. 22 should serve to correct the idea so common among writers of text-books that I- 7 is merely of use as a lemma to Euclid's proof of 1. 8, and therefore may be left out if an alternative proof of that proposition is adopted. Agreeably to his notion that it is from 1. 1 that we must satisfy ourselves that isosceles and scalene triangles actually exist, as well as equilateral triangles, Proclus shows how to draw, lirst a particular isosceles triangle, and then a scalene triangle, by means of the figure of the proposition. To make an isosceles triangle he produces AB in both directions to meet the respective circles in D, E, and then describes circles with A, B as centres and AE, BD as radii respectively. The result is an isosceles triangle with each of two sides double of the third side. To make an isosceles triangle in which the equal sides are not so related to the third side but have any given length would require the use of t. 3 ; and there is no object in treating the question at all in advance of 1. 22. An easier way of satisfying our- selves of the existence of some isosceles triangles would surely be to conceive any two radii of a circle drawn and their extremities joined. There is more point in Proclus' construction of a. scalene triangle, Suppose AC to be a radius of one of the two circles, and D a point on AC lying; in that portion of the circle with centre A which is outside the circle with centre B. Then, joining BD, as in the figure, we have a triangle which obviously has all its sides unequal, that is, a scalene triangle. The above two constructions appear in an-NairlzI's commentary under the name of Heron; Proclus does not mention his source. In addition to the above construction for a scalene triangle (producing a triangle in which the "given" side is greater than one and less than the other of the two remaining sides), Heron has two others showing the other two possible cases, in which the " given " side is (r) less than, (2) greater than, either of the other two sides. ' »44 BOOK I [i. i Proposition 2. To place at a given point (as an extremity) a straight line equal to a given straight line. Let A be the given point, and BC the given straight line. Thus it is required to place at the point A (as an extremity) l a straight line equal to the given straight line BC. From the point A to the point B let the straight line AB be joined ; [Post 1] and on it let the equilateral triangle 10 DAB be constructed. [1. i\ Let the straight lines A£, BF be produced in a straight line with DA, DB; [Post a] with centre B and distance BC let the ■i circle CGH be described ; [Post 3] and again, with centre D and distance DG let the circle GKL be described, [Post. 3] Then, since the point B is the centre of the circle CGH, BC is equal to BG. 20 Again, since the point D is the centre of the circle GKL, DL is equal to DG. And in these DA is equal to DB ; therefore the remainder AL is equal to the remainder BG. [C.N. 3] 2 S But BC was also proved equal to BG ; therefore each of the straight lines AL, BC is equal XoBG. And things which are equal to the same thing are also equal to one another ; [CJK 1] p therefore AL is also equal to BC. Therefore at the given point A the straight line AL is placed equal to the given straight line BC. (Being) what it was required to do. I . (as an extremity}. I have inserted these words because " to place a straight line at a given point " (rftht t£ Bo$4m ffij^ff^) is not quite clear enough, at least in English. II. Let the straight lines AB, BF be produced.,.. It will be observed that in this first application of Postulate a, and again in I. 5, Euclid speaks of the conJinuaiiffft of the straight line as that which is produced in such cases, inlikftXIja&waw and TpoaiKfttffMir#trtar meaning little more than dram'nr straight lines " in a straight line with " the given straight lines. The first place in which Euclid uses phraseology exactly corresponding to ours when I. a] PROPOSITION a *45 speaking of a straight Line being produced ia in I. l6 : "let one side of it, BC t be produced to D " (rp<mt;&ep\1iir8u airoC >t(a irXtupA ij Br * W to A). 13. the remainder AL...the remainder BG. Tbe Greek expressions ate X«r^ ^ A A and XorviJ r£ BH> and the literal translation would be 'Mi (or BG) rtmmning™ bat tbe shade of meaning conveyed by the position of the definite article can hardly be expressed in English. This proposition gives Proclus an opportunity, such as the Greek commentators revelled in, of distinguishing a multitude of cases. After explaining that those theorems and problems are said to have eases which have the same force, though admitting of a number of different figures, and preserve the same method of demonstration while admitting variations of position, and that cases reveal themselves in the etms/rue/tbn, he proceeds to distinguish the cases in this problem arising from the different positions which the given point may occupy relatively to the given straight line. It may be (he says) either (i) outside the line or (a) on the line, and, if (1), it may be either (a) on the line produced or (6) situated obliquely with regard to it ; if (a), it may be either (a) one of the extremities of the line or (A) an intermediate point on it. It will be seen that Proclus' anxiety to subdivide leads him to give a "case," (a) (a), which is useless, since in that "case" we are given what we are required to rind, and there is really no problem to solve. As Savile says, " qui quaerit ad punctum ponere rectam aequaiem rjj fiy rectae, quaerit quod datum est, quod nemo faceret nisi forte msaniat," Proclus gives the construction for (a) (i) following Euclid's way of taking G as the point in .which the circle with centre B intersects DB produced, and then proceeds to " cases," of which there are still more, which result from the different ways of drawing the equilateral triangle and of producing its sides. This last class of "cases" he subdivides into three according as AB is (1) equal to, {a) greater than or (3) less than BC, Here again "case "(i) serves no purpose, since, if AB is equal to BC, the problem is already solved. But Proclus' figures for the other two cases are worth giving, because in one of them the point G is on BD produced beyond D, and in the other it lies on BD itself and there is no need to produce any side of the equilateral triangle. A glance at these figures will show that, if they were used in the proposition, each of them would require a slight modification in the wording (r) of the construction, since BD is in one case produced beyond D instead of B and in the other case not produced at all, (a) of the proof, since BG, instead of being the difference between DG and DB, is in one case the sum of DG and DB and in the other the difference between DB and DG. 14<> BOOK I [i. a, 3 Modern editors generally seem to classify the cases according to the possible variations in the construction rather than according to differences in the data. Thus Lardner, Potts, and Todhunter distinguish eight cases due to the three possible alternatives, (1) that the given point may be joined to either end of the given straight line, {2) that the equilateral triangle may then be described on either side of the joining line, and {3) that the side of the equilateral triangle which is produced may be produced in either direction. (But it should have been observed that, where AB is greater than BC, the third alternative is between producing DB and not producing it at all.) Potts adds that, when the given point lies either on the line or on the line produced, the distinction which arises from joining the two ends of the line with the given point no longer exists, and there are only four cases of the problem (I think he should rather have said solutions). To distinguish a number of cases in this way was foreign to the really classical manner. Thus, as we shall see, Euclid's method is to give one case only, for choice the most difficult, leaving the reader to supply the rest for himself. Where there was a real distinction between cases, sufficient to necessitate a substantial difference in the proof, the practice was to give separate enunciations and proofs altogether as we may see, e.g., from the Cenia and the De section* rationis of Apollonius. Proclus alludes, in conclusion, to the error of those who proposed to solve 1. 2 by describing a circle with the given point as centre and with a distance equal to BC, which, as he says, is a petitio principii. De Morgan puts the matter very clearly {Supplementary Remarks on the first six Books* of Euclid's Elements in the Companion to the Almanac, 1849, p. 6). We should "insist," he says, "here upon the restrictions imposed by the first three postulates, which do not allow a circle to be drawn with a compass-carried distance; suppose the compasses to dose of themselves the moment they cease to touch the paper. These two propositions [1. 2, 3] extend the power of construction to what it would have been if all the usual power of the compasses had been assumed ; they are mysterious to all who do not see that postulate iii does not ask for every use of the compasses." Proposition 3. Given two unequal straight lines, to cut off from the greater a straight line equal to the less. c Let AB, C be the two given un- equal straight lines, and let AB be the greater of them. Thus it is required to cut off from AB the greater a straight line equal to C the less. At the point A let AD be placed equal to the straight line C ; [1. 2] and with centre A and distance AD let the circle DEF be described. [Post 3] I. 3, 4] PROPOSITIONS 2—4 *4T Now, since the point A is the centre of the circle DEF, AE is equal to AD. [Def. 15] But C is also equal to AD. Therefore each of the straight lines AE, C is equal to AD ; so that A E is also equal to C. [C.N. 1] Therefore, given the two straight lines AB, C, from AB the greater AE has been cut off equal to C the less. (Being) what it was required to do. P roc his contrives to make a number of "cases" out of this proposition also, and gives as many as eight figures. But he only produces this variety by practically incorporating the construction of the preceding proposition, instead of assuming it as we are entitled to do. If Prop, 2 is assumed, there is really only one " case " of the present proposition, for Potts distinction between two cases according to the particular extremity of the straight line from which the given length has to be cut off scarcely seems to be worth making. Proposition 4. If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles s will be equal to the remaining angles respectively, namely those which the equal sides subtend. Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE and AC to DF, and the angle BA C equal to the 10 angle EDF. I say that the base BC is also equal to the base EF, the triangle ABC will be equal to the triangle DEF, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that is is, the angle ABC to the angle DEF, and the angle ACB to the angle DEE. For, if the triangle ABC be applied to the triangle DEF, and if the point A be placed ao On the point D and the straight line AB on DE, then the point B will also coincide with E, because AB is equal to DE. 348 BOOK I fjfc 4 »j Again, AB coinciding with DE, the straight line AC will also coincide with DF, because the angle SAC is equal to the angle EDF; hence the point C will also coincide with the point F, because AC is again equal to DF. 30 But B also coincided with E ; hence the base BC will coincide with the base EF. [For if, when B coincides with E and C with F, the base BC does not coincide with the base EF, two straight lines will enclose a space : which is impossible. 35 Therefore the base BC will coincide with EF~\ and will be equal to it. \C.N. 4] Thus the whole triangle ABC will coincide with the whole triangle DEF, and will be equal to it. 40 And the remaining angles will also coincide with the remaining angles and will be equal to them, the angle ABC to the angle DEF, and the angle ACB to the angle DFE, Therefore etc. *S (Being) what it was required to prove. 1 — 3. It is a fact that Euclid's enunciations not infrequently leave something to be desired in point of clearness and precision. Here he speaks of the triangles having *' the angle equal to the angle, namely the angle contained by the equal straight lines " [rrtr yuriar r£ ywif tar}* §xv T h r v*& ruv law iijdtiutv wtpi.txt>M yr l v )i only one of the two angles being described in the latter expression (in the accusative], and a similar expression in the dative being left to be understood of the other angle. It is curious too that, after mentioning two "sides" he speaks of the angles contained by the equal "straight lints" not "sides. It may be that he wished to adhere scrupulously, at the outset, to the phraseology of the definitions, where the angle is the inclination to one another of two lines or straight lints. Similarly in the enunciation of I: £ he speaks of producing the equal " straight lines" as if to keep strictly to the wording of Postulate 1. t. respectively. I agree with Mr H. M. Taylor {Euclid, p. ix) that it is best to abandon the traditional translation of [ ' each to each," which would naturally seem to imply that all the four magnitudes are equal rather than (as the Greek itaripa. 1 tariff does) that one is equal to one and the other to the other. 3. the base. Here we have the word bast used for the first time in the Elements. Proclus explains it (p. 136, 13 — tj) as meaning (1), when no side of a triangle has been mentioned before, the side " which is on a level with the sight " (rV rpbi tq ftipet ntifiirjjr), and (1), when two sides have already been mentioned, the third side. Proclus thus avoids the mistake made by some modern editors who explain the term exclusively with reference to the case where two sides have been mentioned before. That this is an error is proved (t) by the occurrence of .the term in the enunciations of 1. 37 etc. about triangles on the same base and equal bases, (1) by the application of the same term to the bases of parallelograms in h 35 etc. The truth is that the use of the term must have been suggested by the practice of drawing the particular side horizontally, as it were, and the rest of the figure above it. The bait of a figure was therefore spoken of, primarily, in the same sense as the base of anything r. 4] PROPOSITION 4 *49 else. e.g. of a pedestal or column ; but 'vhen, as in ]. 5. two triangles were compared occupying other thun the norma! positions which gave rise to the name, and when two side' had been previously mentioned , the base was as Prod us says, necessarily the third side. 6. subtend. Owvrttteuf br6, " to stretch under," with accusative* 9. the angle BAC. The full Greek expression would be it irro rwp BA, AT vtpuxo^rn yaivtu, " the angle contained by the (straight lines) BA, AC." But it was a common practice of Greek geometers, e.g. of Archimedes and Apollonius (and Euclid too in Books x.— xiij.), to use the abbreviation at BAl' for at BA, AT, "the (straight lines} BA, AC." Thus, on TtpttX'tfni being dropped, the expression would become first ^ iiri rat BAr -yafta, then i M BAP yttrln, and finally i brb BAr, without ywla, as we regularly find it in Euclid. 17. if the triangle be applied to..., 13. coincide. The difference between the technical use of the passive t>apjt4f»T0ai " to be applitd (to)," and of the active t£np^f«> "to coincide (with} has been noticed above (note on Common tfetien 4, pp. 114 — j). J j. [For if, when B coincides... j'>. coincide with EF]. Heiberg (ParaHpumcna i» lid in Hermts, xxxvin., 1003, p. 56] has pointed out, as a conclusive reason for regarding these words as an early interpolation, that the text of an-NairM [Codex Ltidtnsis 300, 1, ed. Besthom- Heiberg, p. 55) does not give the words in this place but after the conclusion q.e.d., which shows that they constitute a scholium only. They were doubt less added b? some commentator who thought it necessary to explain the immediate inference that, since B coincides with E and C with F. the straight line BC coincides with the straight line EF, an inference which really follows from the definition of a straight tine and Post. 1 ; and no doubt the Postulate that "Two straight lines cannot enclose a space" (afterwards placed among the Common Notions) was interpolated at the same time. 44. Therefore etc. Where (as here) Euclid's conclusion merely repeats the enunciation word for word, I shall avoid the repetition and write " Therefore etc" simply. In the note on Common Notion 4 I have already mentioned that Euclid obviously used the method of superposition with reluctance, and I have given, after Veronese for the most part, the reason for holding that that method is not admissible as a theoretical means of proving equality, although it may be of use as a practical test, and may thus furnish an empirical basis on which to found a postulate. Mr Bertrand Russell observes {Principles of Mathematics I. p. 405) that Euclid would have done better to assume 1. 4 as an axiom, as is practically done by Hilbert (Grundtagen der Geometric, p. 9). It may be that Euclid himself was as well aware of the objections to the method as are his modem critics ; but at all events those objections were stated, with almost equal clearness, as early as the middle of the 16th century. Peletarius (Jacques Peletier) has a long note on this proposition (Jn Eudidis Elementa gtometrica demonttratwnunt libri sex, 1557), in which he observes that, if superposition of lines and figures could be assumed as a method of proof, the whole of geometry would be full of such proofs, that it could equally well have been used in 1. 2, 3 (thus in 1. * we could simply have supposed the line taken up and placed at the point), and that in short it is obvious how far removed the method is from the dignity of geometry. The theorem, he adds, is obvious in itself and does not require proof ; although it is introduced as a theorem, it would seem that Euclid intended it rather as a definition than a theorem, " for I cannot think that two angles are equal unless I have a conception of what equality of angles is." Why then did Euclid include the proposition among theorems, instead of placing it among the axioms ? Peletarius makes the best excuse he can, but concludes thus : " Huius itaque propositionis veritatem non aliunde quam a communi iudirio petemus : cogitabimusque figuras figuris superponere, Mechantcum quippiam esse : intelligere verb, id demum esse Mathematicum." Expressed in terms of the modern systems of Congruence-Axioms referred to in the note on Common Notion 4, what Euclid really assumes amounts to the following : (1) On the line DE, there is a point E, on either side of D, such that AB is equal to DE. *5<> BOOK I [1.4 (2) On either side of the ray DE there is a ray DF such that the angle EDF is equal to the angle BAC- It now follows that on DF there is a point ^ such that DF is equal to^C. And lastly (3), we require an axiom from which to infer that the two remaining angles of the triangles are respectively equal and that the bases are equal. I have shown above (pp. 229 — 230) that Hilbert has an axiom stating the equality of the remaining angles simply, but proves the equality of the bases. Another alternative is that of Pasch ( VorUsungen titer neuert Geometric, p. 109) who has the following "Gnmdsatz": If two figures AB and FGH are given (FGH not being contained in a straight length), and AB, FG are congruent, and if a plane surface be laid through A and B, we can specify in this plane surface, produced if necessary, two points C, D, neither more nor less, such that the figures ABC and ABD are congruent with the figure FGH t and the straight line CD has with the straight line AB or with AB produced one point common. I pass to two points of detail in Euclid's proof : (1) The inference that, since B coincides with E, and C with F, the bases of the triangles are wholly coincident rests, as expressly stated, on the impossibility of two straight tines enclosing a space, and therefore presents no difficulty. But (2) most editors seem to have failed to observe that at the very beginning of the proof a much more serious assumption is made without any explanation whatever, namely that, if A be placed on D, and AB on DE, the point B will coincide with £, because AB is equal to DE. That is, the converse of Common Notion 4 is assumed for straight lines. Proem s merely observes, with regard to the converse of this Common Notion, that it is only true in the case of things " of the same form " (i/toeiBi}), which he explains as meaning straight lines, arcs of one and the same circle, and angles " contained by lines similar and similarly situated" (p. 241, 3 — 8^. Savile however saw the difficulty and grappled with it in his note on the Common Notion. After stating that all straight lines with two points common are congruent between them (for otherwise two straight lines would enclose a space), he argues thus. Let there be two straight lines AB, DE, and let A be placed on D, and AB on DE. Then B will coincide with E. For, if not, let B fall somewhere short of E or beyond E ; and in either case it will follow that the less is equal to the greater, which is impossible. Savile seems to assume (and so apparently does Lardner who gives the same proof) that, if the straight lines be " applied," B will fall somewhere on DE or DE produced. But the ground for this assumption should surely be stated ; and it seems to me that it is necessary to use, not Postulate 1 alone, nor Postulate 2 alone, but both, for this purpose (in other words to assume, not only that two straight lines cannot enclose a space, but also that two straight lines cannot have a common segment). For the only safe course is to place A upon D and then turn AB about D until some point on AB intermediate between A and B coincides with some point on DE. In this position AB and DE have two points common. Then Postulate 1 enables us to infer that the straight lines coincide between the two common points, and Postulate 2 that they coincide beyond the second common point towards B and E. Thus the straight lines coincide throughout so far as both extend; and Savile's argument then proves that B coincides with E. I. 5] PROPOSITIONS 4, 5 151 Proposition 5. In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. Let ABC be an isosceles triangle having the side AB 5 equal to the side AC; ' and let the straight lines BD, CE be produced further in a straight line with AB, AC. [Post. 2} 1 say that the angle ABC is equal to the angle ACS, and the angle CBD to the angle BCE. 10 Let a point F be taken at random on BD; from AE the greater let AG be cut off equal to AF the less ; [1. 3] and let the straight lines FC, GB be joined. [Post 1] is Then, since AF is equal to AG and AB to AC, the two sides FA, AC are equal to the two sides GA, AB, respectively ; and they contain a common angle, the angle FAG. 10 Therefore the base FC is equal to the base GB, and the triangle AFC is equal to the triangle AGB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that is, the angle ACF to the angle ABG, 11 and the angle AFC to the angle AGB. [l 4] And, since the whole AF is equal to the whole AG, and in these AB is equal to AC, the remainder BF is equal to the remainder CG. But FC was also proved equal to GB ; v> therefore the two sides BF, FC are equal to the two sides CG, GB respectively ; and the angle BFC is equal to the angle CGB, while the base BC is common to them ; therefore the triangle BFC is also equal to the triangle CGB, (S and the remaining angles will be equal to the remaining 3S* BOOK I [l. 5 angles respectively, namely those which the equal sides subtend ; therefore the angle FBC is equal to the angle GCB, and the angle BCF to the angle CBG. 4° Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG is equal to the angle BCF, the remaining angle ABC is equal to the remaining angle ACB; 45 and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB ; and they are under the base. Therefore etc. Q. e. d. 1. the equal straight lines (meaning the equal sida). Cf. note on the similar expression in Prop. 4, lines 1, 3. 10, Let a point F be taken at random on BD, ettrj^tfu irl rQt BA Tv%6r miitur t& Z, where rvxhr a-rj^fiof means "a chance point/' 17, the two sides FA, AC are equal to the two aides OA, AB respectively, l(So at ZA, AT 0url toTi HA, AB loat flair iuaripa tzartpf. Here, and in numberless later passages, I have inserted the word " sides" for the reason given in the note on 1- r, line 10. It would have been permissible to supply either "straight lines'' or "sides"; but on the whole M sides " seems to be more in accordance with the phraseology of I. 4, 33. the base BC is common to them, i.e., apparently, common to the origin, as the arW-ur in pdtrti a&rww ttotrij can only refer to yuyLa and yuAa preceding. Simson wrote "and the base BC is common to the two triangles BFC, CGB , Todhunter left out these words as being of no use and tending to perplex a beginner. But Euclid evidently chose to quote the conclusion of 1. 4 exactly I the first phrase of that conclusion is that the bases (of the two triangles) are equal, and, as the equal bases are here the same base, Euclid naturally substitutes the worn "common" for '* equal." 48. As " (Being) what it was required to prove " {or " do ") is somewhat long, 1 shall henceforth write the time-honoured "Q. e. D. and "<}. F_ F." for irtp titt ititfit and ortp Wet rotfyrat. According to Proclus (p. 250, 20) the discoverer of the fact that in any isosceles triangle the angles at the base are equal was Thales, who however is said to have spoken of the angles as being similar, and not as being equal. (Cf. ArisL De caelo iv. 4, 311 b 34 n-pos ifiolas ymvlat dxuVcrtu dxpoVcvov where equal angles are meant.) A pre-Euclidean proof of I. 5. One of the most interesting of the passages in Aristotle indicating differences between Euclid's proofs and those with which Aristotle was familiar, in other words, those of the text-books immediately preceding Euclid's, has reference to the theorem of 1. 5. The passage {Anal. Prior. 1. 34, 41 b 13 — 21) is so important that I must quote it in Kill. Aristotle is illustrating the fact that in any syllogism one of the propositions must be affirmative and universal {uttftJAou}. "This," he says, "is better shown in the case of geometrical propositions " {b> toii StaypapiMurtv), e.g. the proposition that the angles at Ike bast of an isosceles triangle are equal. " For let A, B be drawn [i.e. joined] to the centre. I. s] PROPOSITION s *S3 "If, then, we assumed (i) that the angle AC [i.e. A + C] is equal to the angle BD [i.e. B + D\ without asserting generally that the angles of semicircles are equal, and again (a) that the angle C is equal to the angle D without making the further assumption that the two angles of all segments art equal, and if we then inferred, lastly, that, since the whole angles are equal, and equal angles are subtracted from them, the angles which remain, namely E, F, are equal, we should commit a petitio principii, unless we assumed [generally j that, when equals art subtracted from equals, tht remainders are equal.'' The language is noteworthy in some respects. (i ) A, B are said to be drawn (iyy/«Viii) to the centre (of the circle of which the two equal sides are radii) as if A, B were not the angular points but the sides or the radii themselves. (There is a parallel for this in Eucl. iv. 4.) (2) "The angle AC" is the angle which is the sum of A and C, and A means here the angle at A of the isosceles triangle shown in the figure, and afterwards spoken of by Aristotle as E, while C is the " mixed " angle between AB and the circumference of the smaller segment cut off by it. (3) The "angle of a. semicircle" (i.e. the "angle" between the diameter and the circumference, at the extremity of the diameter) and the " angle of a segment" appear in Euclid tn. 16 and 111. Def. 7 respectively, obviously as survivals from earlier text-books. But the most significant facts to be gathered from the extract are that in the text-books which preceded Euclid's " mixed " angles played a much more important part than they do with Euclid, and, in particular, that at least two propositions concerning such angles appeared quite at the beginning, namely the propositions that the (mixed) angles of semicircles art equal and that the two (mixed) angles of any segment of a circle art equal. The wording of the first of the two propositions is vague, but it does not necessarily mean more than that the two (mixed) angles in one semicircle are equal, and I know of no evidence going to show that it asserts that the angle of any one semicircle is equal to the angle of any other semicircle (of different size). It is quoted in the same form, " because the angles of semicircles are equal," in the Latin translation from the Arabic of Heron's Catopirica, Prop. 9 (Heron, Vol. 11., Teubner, p. 334), but it is only inferred that the different radii of one circle make equal "angles" with the circumference; and in the similar proposition of the Pseudo-Euclidean Catoptriea (Euclid, Vol. vn., p. 394) angles of the same sort in one circle are said to be equal " because they are (angles) of a semicircle." Therefore the first of the two propositions may be only a particular case of the second. But it is remarkable enough that the second proposition (that the two "angles of" any segment of a circle art equal) should, in earlier text-books, have been placed before the theorem of Eucl. 1. 5. We can hardly suppose it to have been proved otherwise than by the superposition of the semicircles into which the circle is divided by the diameter which bisects at right angles the base of the segment; and no doubt the proof would be closely connected with that of Thales' other proposition that any diameter of a circle bisects it, which must also (as Proclus indicates) have been proved by superposing one of the two parts upon the other. It is a natural inference from the passage of Aristotle that Euclid's proof of *54 BOOK I [i. s i. 5 was his own, and it would thus appear that his innovations as regards order of propositions and methods of proof began at the very threshold of the subject. Proof without producing the sides. In this proof, given by Proclus (pp. 148, 21—249, J 9)> & an ^ E are ta ^en on AS, AC, instead of on AB, AC produced, so that AD, AE&te equal. The method of proof is of course exactly like Euclid's, but it does not establish the equality of the angles beyond the base as well. Pappus* proof. Proclus (pp. 249, 20 — 250, 1 2) says that Pappus proved the theorem in a still shorter manner without the help of any construction whatever. This very interesting proof is given as follows : " Let ABC be an isosceles triangle, and AB equal to AC. Let us conceive this one triangle as two triangles, and let us argue in this way. Since AB is equal to AC, and AC to AB, the two sides AB, AC are equal to the two sides AC, AB. And the angle BA C is equal to the angle CAB, for it is the same. Therefore all the corresponding parts (in the triangles) are equal, namely BC to BC, the triangle ABC to the triangle ABC (i.e. ACB), the angle ABC to the angle ACB, and the angle ACB to the angle ABC, (for these are the angles subtended by the equal sides AB, A C. Therefore in isosceles triangles the angles at the base are equal." This will no doubt be recognised as the foundation of the alternative proof frequently given by modern editors, though they do not refer to Pappus. But they state the proof in a different form, the common method being to suppose the triangle to be taken up, turned over, and placed again upon itself, after which the same considerations of congruence as those used by Euclid in 1. 4 are used over again. There is the obvious difficulty that it supposes the triangle to be taken up and at the same time to remain where it is, (Cf. Dodgson's humorous remark upon this, Euclid and Ail modern Rivals, p. 47.) Whatever we may say in justification of the proceeding (e.g. that the triangle may be supposed to leave a tract), it is really equivalent to assuming the construction (hypothetical, if you will) of another triangle equal in all respects to the given triangle ; and such an assumption is not in accordance with Euclid's principles and practice. It seems to me that the form given to the proof by Pappus himself is by far the best, for the reasons (i) that it assumes no construction of a second triangle, real or hypothetical, (2) that it avoids the distinct awkwardness involved by a proof which, instead of merely quoting and applying the result of a previous proposition, repeats, with reference to a new set of data, the process by which that result was established. If it is asked how we are to realise Pappus' idea of two triangles, surely we may answer that we keep to one triangle and merely view it in two aspects. If it were a question of helping a beginner to understand this, we might say that one triangle is the triangle i. 5, 6] PROPOSITIONS 5, 6 355 looked at in front and that the other triangle is the same triangle looked at from behind ; but even this is not really necessary. Pappus' proof, of course, does not include the proof of the second part of the proposition about the angles under the base, and we should still have to establish this much in the same way as Euclid does. Purpose of the second part of the theorem. An interesting question arises as to the reason for Euclid's insertion of the second part, to which, it will be observed, the converse proposition 1. 6 has nothing corresponding. As a matter of fact, it is not necessary for any subsequent demonstration that is to be found in the original text of Euclid, but only for the interpolated second case of 1. 7 ; and it was perhaps not unnatural that the undoubted genuineness of the second part of 1. 5 convinced many editors that the second case of 1. 7 must necessarily be Euclid's also. Proctus' explanation, which must apparently be the right one, is that the second part of 1. 5 was inserted for the purpose of fore-arming the learner against a possible objection (frimturif), as it was technically called, which might be raised to 1. 7 as given in the text, with one case only. The objection would, as we have seen, take the specific ground that, as demonstrated, the theorem was not conclusive, since it did not cover all possible cases. From this point of view, the second part of 1. 5 is useful not only for 1. 7 but, according to Proclus, for 1. 9 also. Simson does not seem to have grasped Proclus' meaning, for he says : " And Proclus acknowledges, that the second part of Prop. 5 was added upon account of Prop. 7 but gives a ridiculous reason for it, 'that it might afford an answer to objections made against the 7th,' as if the case of the 7th which is left out were, as he expressly makes it, an objection against the proposition itself." Proposition 6. If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another. Let ABC be a triangle having the angle ABC equal to the angle ACS', I say that the side AB is also equal to the side AC. For, if AB is unequal to AC, one of them is greater. Let AB be greater; and from AB the greater let DB be cut off equal to AC the less ; let DC be joined. Then, since DB is equal to AC, and BC is common, the two sides DB, BC are equal to the two sides AC, CB respectively ; *$6 BOOK I [i. 6 and the angle DBC is equal to the angle ACB ; therefore the base DC is equal to the base AB, and the triangle DBC will be equal to the triangle ACB, the less to the greater : which is absurd. Therefore AB is not unequal to AC; it is therefore equal to it. Therefore etc Q. E. D. Euclid assumes that, because D is between A and B, the triangle DBC is less than the triangle ABC. Some postulate is necessary to justify this tacit assumption; considering an angle less than two right angles, say the angle ACB in the figure of the proposition, as a cluster of rays issuing from C and bounded by the rays CA f CB, and joining AB (where A, B are any two points on CA, CB respectively), we see that to each successive ray taken in the direction from CA to CB there corresponds one point on AB in which the said ray intersects AB, and that all the points on AB taken in order from A to B correspond uni vocally to all the rays taken in order from CA to CB, each point namely to the ray intersecting AB in the point. We have here used, for the first time in the Elements, the method of redact in ad absurdum, as to which I would refer to the section above (pp. 136, 140) dealing with this among other technical terms. This proposition also, being the converse of the preceding proposition, brings us to the subject of Geometrical Conversion. This must of course be distinguished from the logical conversion of a proposition. Thus, from the proposition that alt isosceles triangles have the angles opposite to the equal sides equal, logical conversion would only enable us to conclude that some triangles with two angles equal are isosceles. Thus 1. 6 is the geometrical, but not the logical, converse of 1. 5. On the other hand, as De Morgan points out (Companion to the Almanac, 1849, p. 7), 1. 6 is a purely logical deduction from t. 5 and 1. 18 taken together, as is 1. 19 also. For the general argument see the note on 1. 19. For the present proposition it is enough to state the matter thus. Let X denote the class of triangles which have the two sides other than the base equal, Y the class of triangles which have the base angles equal ; then we may call aan-X the class of triangles having the sides other than the base unequal non- Y the class of triangles having the base angles unequal. Thus we have aii x is r, [1. 5] All aon-X is non-K; [1. 18) and it is a purely logical deduction that All Y is X. [1. 6] According to Proclus (p. 252, 5 sqq.) two forms of geometrical conversion were distinguished. (1) The leading form {rpo^yovji.iyin). the conversion par excellence (rj xvpum t. 6] PROPOSITION 6 457 awKrrpo^itf), is the complete or simple conversion in which the hypothesis and the conclusion of a theorem change places exactly, the conclusion of the theorem being the hypothesis of the converse theorem, which again establishes, as its conclusion, the hypothesis of the original theorem. The relation between the first part of 1. 5 and 1. 6 is of this character. In the former the hypothesis is that two sides of a triangle are equal and the conclusion is that the angles at the base are equal, while the converse {1. 6) starts from the hypothesis that two angles are equal and proves that the sides subtending them are equal. (2) The other form of conversion, which we may call partial, is seen in cases where a theorem starts from two or more hypotheses combined into one enunciation and leads to a certain conclusion, after which the converse theorem takes this conclusion in substitution for one of the hypotheses of the original theorem and from the said conclusion along with the rest of the original hypotheses obtains, as its conclusion, the omitted hypothesis of the original theorem, r, 8 is in this sense a converse proposition to 1. 4 ; for 1. 4 takes as hypotheses ( 1 ) that two sides in two triangles are respectively equal, (a) that the included angles are equal, and proves (3) that the bases are equal, while 1. 8 takes {1) and (3) as hypotheses and proves (2) as its conclusion. It is clear that a conversion of the leading type must be unique, while there may be many partial conversions of a theorem according to the number of hypotheses from which it starts. Further, of convertible theorems, those which took as their hypothesis the genus and proved a, property were distinguished as the leading theorems (rponfyovptva), while those which started from the property as hypothesis and described, as the conclusion, the genus possessing that property were the converse theorems. 1. 5 is thus the leading theorem and [. 6 its converse, since the genus is in this case taken to be the isosceles triangle. Converse of second part of I. 5. Why, asks Proclus, did not Euclid convert the second part of I. 5 as well ? He suggests, properly enough, two reasons: (1) that the second part of 1. 5 itself is not wanted for any proof occurring in the original text, but is only put in to enable objections to the existing form of later propositions to be met, whereas the converse is not even wanted for this purpose ; (2) that the converse could be deduced from t. 6, if wanted, at any time after we have passed 1. 13, which can be used to prove that, if the angles formed by producing two sides of a triangle beyond the base are equal, the base angles themselves are equal. Proclus adds a proof of the converse of the second part of 1. 5. i.e. of the proposition that, if the angles formed by producing two sides of_ a triangle beyond the base are equal, the triangle a is isosceles ; but it runs to some length and then only „ A effects a reduction to the theorem of 1. 6 as we have it. AA As the result of this should hardly be assumed, a better / \\ proof would be an independent one adapting Euclid's / ^ own method in 1. 6. Thus, with the construction of 1. 5, l"~^~~^\ we first prove by means of 1. 4 that the triangles BFC, /-^"^*^~^\ CGB are equal in all respects, and therefore that FC is y~ "^^ equal to GB, and the angle BFC equal to the angle CGB. D g Then we have to prove that AF, AG are equal. If they are not, let AF be the greater, and from FA cut off FH equal to GA. Join CH. i S 8 BOOK I [i.«,T Then we have, in the two triangles HFC, AGB, two sides HF, FC equal to two sides AG, GB and the angle HFC equal to the angle AGB. Therefore (l 4) the triangles HFC, AGB are equal. But the triangles BFC, CGB are also equal Therefore {if we take away these equals respectively) the triangles HBC, ACB are equal: which is impossible. Therefore AF, AG are not unequal. Hence AF'm equal to AG and, if we subtract the equals BF, CG respec- tively, AB is equal to A C. This proof is found in the commentary of an-Nairlzi (ed. Besthom-Heiberg, p. 61 ; ed. Curtze, p. 50). Alternative proofs of I. 6. Todhunter points out that I. 6, not being wanted till II. 4, could be postponed till later and proved by means of i. 26. Bisect the angle BAC by a straight line meeting the base at D. Then the triangles ABD, A CD are equal in all respects. Another method depending on 1. 26 is given by an-Nairlzi after that proposition. Measure equal lengths BD, CE along the sides BA, CA. Join BE, CD. Then [1. 4] the triangles DBC, ECB are equal in all respects ; therefore EB, DC are equal, and the angles BEC, CDB are equal. The supplements of the latter angles are equal [1. 13], and hence the triangles ABE, A CD have two angles equal respectively and the side BE equal to the side CD. Therefore [1. 26] AB is equal to AC. P HO POSITION 7. Given two straight lines constructed on a straight line [from its extremities) and meeting in a point, there cannot be constructed on the same straight line {from its extremities), and on the same side of it, two other straight lines meeting in 5 another point and equal to the former two respectively, namely each to that which has the same extremity with it. For, if possible, given two straight lines AC, CB con- structed on the straight line AB and meeting at the point C, let two other straight lines 10 AD, DB be constructed on the same straight line AB, on the same side of it, meeting in another point D and equal to the former two respectively, namely each to that which has the same extremity with it, so that CA is 15 equal to DA which has the same extremity A with it, and I. 7] PROPOSITIONS 6, 7 259 CB to DB which has the same extremity B with it ; and let CD be joined. Then, since AC is equal to AD, the angle A CD is also equal to the angle ADC; [1. 5] 20 therefore the angle ADC is greater than the angle DCB ; therefore the angle CDB is much greater than the angle DCB. Again, since CB is equal to DB, the angle CDB is also equal to the angle DCB. 25 But it was also proved much greater than it : which is impossible. Therefore etc. Q. E. D. 1 — 6. In an English translation of the enunciation of this proposition it is absolutely necessary, in order to make it intelligible, to insert some words which are not in the Greek. The reason is partly that the Greek enunciation is itself very elliptical, and partly that some words used in it conveyed more meaning than the corresponding words in English do. Particularly is this the case with oC evaratrboorrat iri "there shall not be constructed upon," since evrUraaSat. is the regular word for constructing a triangle in particular. Tbus a Greek would easily understand avaraB-fyrarrat iri as meaning the construction of two lines forming a triangle on a given straight line as base; whereaa.ro "construct two straight lines on a straight line " is not in English sufficiently definite unless we explain that they are drawn from the ends of the straight line to meet at a point. I have had the less hesitation in putting in the words "from its extremities" because they are actually used by Euclid in the somewhat similar enunciation of 1. at. How impossible a literal translation into English is, if it is to convey the meaning of the enunciation intelligibly, will be clear from the following attempt to render literally: "On the same straight line there shall not be constructed two other straight lines equal, each to each, to the same two straight lines, (terminating) at different points on the same side, having the same extremities as the original straight lines " (irl riji a*rj)i eiBctai S60 raft airaii tiStlmt AXXor ivo evSttat focu inar4pa ixarlpa ov ewTaBfaairrai vpdr dWt^j rtai AXX^I OTjfuU^ i-wl f A ovrd pift r« a*"k ripara (x'oeai roll it Apx*)' tiltlaa). The reason why Euclid allowed himself to use, in this enunciation, language apparently so obscure is no doubt that the phraseology was traditional and therefore, vague as it was, had a conventional meaning which the contemporary geometer well understood. This is proved, I think, by the occurrence in Aristotle (Meteorologica 111, 5, 376 a j sqq.) of the very same, evidently technical, expressions. Aristotle is there alluding to the theorem given by Eutocius from Apollonius' Plane Loci to the effect that, if H, K be two fixed points and M such a variable point that the ratio of MH to MK is a given ratio (not one of equality), the locus of M is a circle. (For an account of this theorem see note on vt. 3 below.) Now Aristotle says "The lines drawn up from H, K in this ratio cannot be constructed to two different points of the semicircle A " (of ttr iri tCh HK inayb/urai ypawial tr roirip rif \t>yy uLr ffvffTad^aovTat rov £q> y A i^uxurXfou rpbt AWo teal A\\o a-rjfLtiov). If a paraphrase is allowed instead of a translation adhering as closely as possible to the original, Simson's is the best that could be found, since the fact that the straight lines form triangles on the same base is really conveyed in the Greek. Simson's enunciation is, Upon the same base, and on the same side of i(, there cannot lie two triangles line have their sides which are terminated in one extremity of the iate equal to one another, and liiewue these which art terminated at the other extremity. Th. Taylor (the translator of Proclus) attacks Simson's alteration as "indiscreet" and as detracting from the beauty and accuracy of Euclid's enunciation which are enlarged upon by Proclus in his commentary. Yet, when Taylor says "Whatever difficulty learners may find in conceiving this proposition abstractedly is easily removed by its exposition in the figure," he really gives his case away. The fact is that Taylor, always enthusiastic over his author, was nettled by Simson's slighting remarks on Proclus' comments on the proposition. Simson had said, with reference to Proclus' explanation of the bearing of the second part of 1. % on 1. 7, that it was not "worth while j6o BOOK t [i. 7 to relate his [rifles at full length," to which Taylor retorts "But Mr Simson was no philosopher ; and therefore the greatest part of these Commentaries must be considered by him as trifles, from the want of a philosophic genius to comprehend their meaning, and a taste superior to that of a mere mathematician, to discover their beauty and elegance." 10. It would be natural to insert here the step "but the angle ACD is greater than the angle BCD. [C. N. 5.3" tl, much greater, literally ''greater by much" (roXX^i fielfav). Simson and those who follow him translate : " much mart then is the angle BDC greater than the angle BCD," but the Greek for this would have to be toXXv (or iroXii) jiaX \i r tm...JU(JW>. troXX^ ^aXXof, however, though used by Apotlonius, is not, apparently, found in Euclid or Archimedes. Just as in I. 6 we need a Postulate to justify theoretically the statement that CD falls within the angle ACB, so that the triangle DBC is less than the triangle ABC, so here we need Postulates which shall satisfy us as to the relative positions of CA, CB, CD on the one hand and of DC, DA, DB on the other, in order that we may be able to infer that the angle BDC is greater than the angle ADC, and the angle ACD greater than the angle BCD, De Morgan {sp. cit. p. 7) observes that 1. 7 would be made easy to beginners if they were first familiarised, as a common notion, with " if two magnitudes be equal, any magnitude greater than the one is greater than any magnitude less than the other." I doubt however whether a beginner would follow this easily ; perhaps it would be more easily apprehended tn the form "if any magnitude A is greater than a magnitude B, the magnitude A is greater than any magnitude equal to B, and (a fortiori) greater than any magnitude less than B." It has been mentioned already (note on 1. 5) that the second case of 1. 7 given by Simson and in our text-books generally is not in the original text (the omission being in accordance with Euclid's general practice of giving only one case, and that the most difficult, and leaving the others to be worked out by the reader for himself). The second case is given by Proclus as the answer to a possible objection to Euclid's proposition, which should assert that the proposition is not proved to be universally true, since the proof given does not cover all possible cases. Here the objector is supposed to contend that what Euclid declares to be impossible may still be possible if one pair of lines lie wholly within the other pair of lines; and the second part of 1. 5 enables the objection to be refuted. If possible, let AD, DB be entirely within the triangle formed by AC, CB with AB, and let AC be equal to AD and BC to BD. Join CD, and produce AC, AD to E and F. Then, since AC is equal to AD, o( the triangle ACD is isosceles, and the angles ECD, FDC under the base are equal. But the angle ECD is greater than the angle BCD , therefore the angle FDC is also greater than the angle BCD. Therefore the angle BDC is greater by far than the angle BCD. Again, since DB is equal to CB, the angles at the base of the triangle BDC are equal, [1. 5] that is, the angle BDC is equal to the angle BCD. Therefore the same angle BDC is both greater than and equal to the angle BCD: which is impossible. The case in which D falls on AC or BC does not require proof, i. 7. 8] PROPOSITIONS 7. 8 *6i I have already referred (note on 1. r) to the mistake made by those editors who regard I. 7 as being of no use except to prove 1. 8. What 1. 7 proves is that if, in addition to the base of a triangle, the length of the side terminating at each extremity of the base is given, only one triangle satisfying these conditions can be constructed on one and the same side of the given base. Hence not only does 1. 7 enable us to prove 1. 8, but it supplements 1. 1 and 1. 22 by showing that the constructions of those propositions give one triangle only on one and the same side of the base. But for [. 7 this could not be proved except by anticipating in, 10, of which therefore 1. 7 is the equivalent for Book 1. purposes. Dodgson (Etttfid and his modern Rivals, pp. 194 — 5) puts it in another way. " It [l. 7] shows that, of airplane figures that can be made by hingeing rods together, the /Aree-sided ones (and these only) are rigid (which is another way of stating the fact that there cannot be two such figures on the same base). This is analogous to the fact, in relation to solids contained by plane surfaces hinged together, that any such solid is rigid, there being no maximum number of sides. And there is a close analogy between 1. 7, 8 and in. 23, 24. These analogies give to geometry much of its beauty, and I think that they ought not to be lost sight of." It will therefore be apparent how ill-advised are those editors who eliminate 1. 7 altogether and rely on Philo's proof for 1. 8. Proclus, it may be added, gives (pp. 2 68, 19 — 269, 10) another explanation of the retention of I. 7, notwithstanding that it was apparently only required for 1. 8. It was said that astronomers used it to prove that three successive eclipses could not occur at equal intervals of time, i.e. that the third could not follow the second at the same interval as the second followed the first ; and it was argued that Euclid had an eye to this astronomical application of the proposition. But, as we have seen, there are other grounds for retaining the proposition which are quite sufficient of themselves. Proposition 8. If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines. s Let ABC, DBF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE, and AC to DF\ and let them have the base BC equal 10 to the base EF ; I say that the angle BAC is also equal to the angle EDF. For, if the triangle ABC be applied to the triangle DEF, and if the point B be placed on 15 the point E and the straight line BC on EF, the point C will also coincide with F, because BC is equal to EF. 2 62 BOOK I [l8 Then, BC coinciding with EF, BA, AC will also coincide with ED, DF; ao for, if the base BC coincides with the base EF, and the sides BA, AC do not coincide with ED, DF but fall beside them as EG, GF, then, given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there will »5 have been constructed on the same straight line (from its extremities), and on the same side of It, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it 30 But they cannot be so constructed. [1. 7} Therefore it is not possible that, if the base BC be applied to the base EF, the sides BA, AC should not coincide with ED, DF; they will therefore coincide, 35 so that the angle BAC will also coincide with the angle EDF t and will be equal to it. If therefore etc. q. e. d. 19. BA, AC. The text has here " BA, CA." SI, fall be aide them. The Greek has the future, i-apoXXdfoLvf. TapaXk&m* means "to pass by without touching," "to miss" or "to deviate." As pointed out above (p. 157) 1, 3 is a par tied converse of t. 4. It is to be observed that in I. 8 Euclid is satisfied with proving the equality of the vertical angles and does not, as in 1. 4, add that the triangles are equal, and the remaining angles are equal respectively. The reason is no doubt (as pointed out by Proclus and by Savile after him) that, when once the vertical angles are proved equal, the rest follows from 1. 4, and there is no object in proving again what has been proved already. Aristotle has an allusion to the theorem of this proposition in Me&orokgka in. 3, 373 a 5 — 16. He is speaking of the rainbow and observes that, if equal rays be reflected from one and the same point to one and the same point, the points at which reflection takes place are on the circumference of a circle. "For let the broken lines ACB, AFB, ADB be all reflected from the point A to the point B (in such a way that) AC, AF, AD are all equal to one another, and the lines {terminating) at B, i.e. CB, FB, DB, are likewise all equal ; and let AEB be joined. It follows that the triangles art equal; for they are upon the equal (base) AEB." Heiberg {Mathtmatisehes tu Aristolelts, p. 18) thinks that the form of the conclusion quoted is an indication that in the corresponding proposition tc. Eucl. 1. 8, as it lay before Aristotle, it was maintained that the triangles were equal, and not only the angles, and "we see here therefore, in a clear example, how the stones of the ancient fabric were recut for the rigid structure of his i 8] PROPOSITION 8 a6j Elements. " I do not, however, think that this inference from Aristotle's language as to the form of the pre- Euclidean proposition is safe. Thus if we, nowadays, were arguing from the data in the passage of Aristotle, we should doubtless infer directly that the triangles are equal in all respects, quoting I 8 alone. Besides, Aristotle's language is rather careless, as the next sentences of the same passage show. "Let perpendiculars," he says, " be drawn to AEB from the angles, CE from C, FE from ^and DE from D. These, then, are equal ; for they are all in equal triangles, and in one plane ; for all of them are perpendicular to AEB, and they meet at one point E. There- fore the (line) drawn (through C, F, D) will be a circle, and its centre (will be) E." Aristotle should obviously have proved that the three perpendiculars will meet at one point E cm AEB before he spoke of drawing the perpendiculars CE, FE, DE. This of course follows from their being "in equal triangles" {by means of EucL i. 26); and then, from the fact that tbe perpendiculars meet at one point on AB, it can be inferred that all three are in one plane. Philo's proof of I. 8. This alternative proof avoids the use of 1. 7, and it is elegant ; but it is inconvenient in one respect, since three cases have to be distinguished. Proctus gives the proof in the following order (pp. 266, 15 — 168, 14). I-et ABC, DEF be two triangles having the sides AB, A C equal to the sides DE, DE respectively, and the base BC equal to the base EF. Let the triangle ABC be applied to the triangle DEF, so that B is placed on E and BC on EF, but so that A falls on the opposite side of EF from D, taking the position G. Then C will coincide with F, since BC is equal to EF. Now FG will either be in a straight line with DF, or make an angle with it, and in the tatter case the angle will either be interior (™ro to Ivtos) to the figure or exterior (no.™ to Item). I. Let FG be in a straight line with DF. Then, since DE is equal to EG-, and DFG is a straight line, DEG is an isosceles triangle, and the angle at D is equal to the angle at G. ['■ 5]. II. Let DF, FG form an angle interior to the figure. Let DG be joined. Then, since DE, EG are equal, the angle EDG is equal to the angle EGD. Again, since DF is equal to FG, the angle FDG is equal to the angle FGD. Therefore, by addition, the whole angle EDF is equal to the whole angle EGF. A i6 4 BOOK I [i. 8, 9 III. Let DF, FG form an angle ex/trier to the figure. Let DG be joined. The proof proceeds as in the last case, except that subtraction takes the place of addition, and the remaining angle EDF is equal to the remaining angle ECF. Therefore in all three cases the angle EDF is equal to the angle EGF, that is, to the angle BAC. It wrill be observed that, in accordance with the practice of the Greek geometers in not recognising as an "angle" any angle not less than two right angles, the re-entrant angle of the quadrilateral JDEGF'm. ignored and the angle DFG is said to be outside the figure. Proposition 9. To bisect a given rectilineal angle. Let the angle BAC be the given rectilineal angle. Thus it is required to bisect it. Let a point D be taken at random on AB ; let AE be cut off from AC equal to AD ; [1. 3] let DE be joined, and on DE let the equilateral triangle DEF be constructed ; let AF be joined. I say that the angle BAC has been bisected by the straight line AF. For, since AD is equal to AE, and AF is common, the two sides DA, AF are equal to the two sides EA, AF respectively. And the base DF is equal to the base EF; therefore the angle DAF is equal to the angle EAF. [.. 8] Therefore the given rectilineal angle BAC has been bisected by the straight line AF. q, e. f. It will be observed from the translation of this proposition that Euclid does not say, in his description of the construction, that the equilateral triangle should be constructed on the side of HE opposite to A ; he leaves this to be inferred from his figure. There is no particular value in Proclus' explanation as to how we should proceed in case any one should assert that he could not recognise the existence of any space below DE. He supposes, then, the equilateral triangle described on the side of DE towards A, and hence has to consider three cases according as the vertex of the equilateral triangle falls on A, above A or below it. The second and third cases do not d'ffer >.«] PROPOSITIONS 8, 9 265 substantially from Euclid's. In the first case, where ADE is the. equilateral triangle constructed on DE, take any point F or\ AD, and from AE cut off AG equal to AF. Join DG, EF meeting in H\ and join AH. Then AH is the bisector required. Proclus also answers the possible objection that might be raised to Euclid's proof on the ground that it assumes that, if the equilateral triangle be described on the side of DE opposite to A, its vertex .f will lie within the angle BAC. The objector is supposed to argue that this is not necessary, but that F might fall either on one of the lines forming the angle or outside it altogether. The two cases are disposed of thus. Suppose Fxx> fall as shown in the two figures below respectively. Then, since FD is equal to FE, the angle FDE is equal to the angle FED. Therefore the angle CED is greater than the angle FDE ; and, in the second figure, a fortiori, the angle CED is greater than the angle BDE. But, since ADE is an isosceles triangle, and the equal sides are produced, the angles under the base are equal, i.e., the angle CED is equal to the angle BDE. But the angle CED was proved greater : which is impossible. Here then is the second case in which, in Proclus' view, the second part of i. 5 is useful for refuting objections. On this proposition Proclus takes occasion (p. 27 r, rj — 19) to emphasize the fact that the given angle must be rectilineal, since the bisection of any sort of angle (including angles made by curves with one another or with straight lines) is not matter for an elementary treatise, besides which it is questionable whether such bisection is always possible. "Thus it is difficult to say whether it is possible to bisect the so-called horn-like angle " (formed by the circumference of a circle and a tangent to it). Trisection of an angle. Further it is here that Proclus gives us his valuable historical note about the trisection of any acute angle, which (as well as the division of an angle in any given ratio) requires resort to other curves than circles, i.e. curves of the species which, after Geminus, he calls "mixed." "This," he says (p. 372, 1 — 12), "is shown by those who have set themselves the task of trisecting such a given rectilineal angle. For Nicomedes trisected any rectilineal angle by means of the conchoidal lines, the origin, order, and properties of which he has handed down to us, being himself the discoverer of their peculiarity. Others have done the same thing by means of the quadratrices of Hippias and Nicomedes, thereby again using 'mixed' curves. But others, starting from the Archimedean spirals, cut a given rectilineal angle in a given ratio." i66 BOOK I [1.9 (a) T riscction by means of the conchoid. I have already spoken of the conchoid of Nicomedes {note on Def. t, pp. 160 — i); it remains to show how it could be used for trisecting an angle. Pappus explains this (iv. pp. 274 — 5) as follows. Let ABC be the given acute angle, and from any point A in AB draw A C perpendicular to BC. B O Complete the parallelogram FBCA and produce FA to a point E such that, if BE be joined, BE intercepts between AC and AE a length DE equal to twice AB. I say that the angle EBC is one-third of the angle ABC. For, joining A to G, the middle point of DE, we have the three straight lines AG, DG, EG equal, and the angle AGO is double of the angle A ED or EBC. But DE is double of AB ; therefore AG, which is equal to DG, is equal to AB. Hence the angle AGD is equal to the angle ABG. Therefore the angle ABD is also double of the angle EBC; so that the angle EBC is one-third of the angle ABC. So far Pappus, who reduces the construction to the drawing of BE so that DE shall be equal to twice AB. This is what the conchoid constructed with B as pole, AC 'as directrix, and distance equal to twice AB enables us to do ; for that conchoid cuts AE in the required point E. (6) Use of the quadratrix. The plural quadratrices in the above passage is a Hellenism for the singular quadratrix, which was a curve discovered by Hippias of El is about 420 B.C. According to Prod us (p. 356, 11) Hippias proved its properties; and we are told (1) in the passage quoted above that Nicomedes also investigated it and that it was used for trisecting an angle, and (2) by Pappus (iv. pp. 350, 33— 353, 4) that it was used by Dinostratus and Nicomedes and some more recent writers for squaring the circle, whence its name. It is described thus (Pappus iv. p. 352). Suppose that ABCD is a square and BED a quadrant of a circle with centre A. Suppose (1) that a radius of the circle moves uniformly about A from the position AB to the position AD, and (*) that in the same time the line BC moves uniformly, always parallel to itself, and with its extremity B moving along BA, from the position BC to the position AD. Then the radius AE and the moving line BC determine at any instant by their intersection a point F, The locus of F'\s the quadratrix. I. 9, 10] PROPOSITIONS o, 10 167 The property of the curve is that, if F is any point, the arc BED is to the arc ED as AB is to FH. In other words, if ^ is the angle FAD, p the radius vector AFa.nd a the side of the square, (p sin $)/« = $/Jt. Now the angle EAD can not only be trisected but divided in any given ratio by means of the quadratrix (Pappus iv. p. 386). For let FJfbe divided at JCin the given ratio. Draw KL parallel to AD, meeting the curve in L ; join AL and produce it to meet the circle in N. Then the angles EAN, NAD are in the ratio of FK to KH, as is easily proved. (e) Use of the spiral of Archimedes. The trisect ion of an angle, or the division of an angle in any ratio, by means or the spiral of Archimedes is of course an equally simple matter. Suppose any angle included between the two radii vectores OA and OB of the spiral, and let it be required to cut the angle AOB in a given ratio. Since the radius vector increases proportionally with the angle described by the vector which generates the curve (reckoned from the original position of the vector coinciding with the initial line to the particular position assumed), we have only to take the radius vector OB (the greater of the two OA, OB), mark off OC along it equal to OA, cut CB in the given ratio (at D say}, and then draw the circle with centre and radius OD cutting the spiral in E. Then OE will divide the angle AOB in the required manner. Proposition 10. To bisect a given finite straight line. Let AB be the given finite straight line. Thus it is required to bisect the finite straight line AB. Let the equilateral triangle ABC be constructed on it, [1. r] and let the angle ACB be bisected by the straight line CD ; ft, 9] I say that the straight line AB has been bisected at the point D. . For, since AC is equal to CB, and CD is common, the two sides A C, CD are equal to the two sides BC, CD respectively ; and the angle A CD is equal to the angle BCD ; therefore the base AD is equal to the base BD. [1. 4] Therefore the given finite straight line AB has been bisected at D. q. e, f. 268 BOOK I [>■ A poll on i us, we are told (P rod us, pp. 279, 16—280, 4), bisected a straight line AB by a construction tike that of 1. 1. With centres A, B, and radii AB, BA respec- tively, two circles are described, intersecting in C, p. Joining CD, AC, CB, AD, DB, Apoi- lonius proves in two steps that CD bisects AB. (1) Since, in the triangles A CD, BCD, two sides AC, CD are equal to two sides BC, CD, and the bases AD, BD are equal, the angle A CD is equal to the angle BCD. [1.8] (2) The latter angles being equal, and AC being equal to CB, while CE is common, the equality of AE, EB follows by I. 4. The objection to this proof is that, instead of assuming the bisection of the angle ACB, as already effected by 1. 9, Apollo nius goes a step further back and embodies a construction for bisecting the angle. That is, he unnecessarily does over again what has been done before, which is open to objection from a theoretical point of view. Proclus (pp. 277, 25 — 279, 4) warns us against being moved by this proposition to conclude that geometers assumed, as a preliminary hypothesis, that a line is not made up of indivisible parts (1$ AjttpAr). This might be argued thus. If a line is made up of indivisibles, there must be in a finite line either an odd or an even number of them. If the number were odd, it would be necessary in order to bisect the line to bisect an indivisible (the odd one). In that case therefore it would not be possible to bisect a straight line, if it is a magnitude made up of indivisibles. But, if it is not so made up, the straight line can be divided ad infinitum or without limit (hf irtipov faatpttTai). Hence it was argued (^acrtV), says Proclus, that the divisibility of magnitudes without limit was admitted and assumed as a geometrical principle. To this he replies, following Geminus, that geometers did indeed assume, by way of a common notion, that a continuous magnitude, i.e. a magnitude consisting of parts connected together (owrjfijtow), is divisible (SuuptTov). But infinite divisibility was not assumed by them ; it was proved by means of the first principles applicable to the case. "For when," he says, " they prove that the incommensurable exists among magnitudes, and that it is not all things that are commensurable with one another, what else will any one say that they prove but that every magnitude can be divided for ever, and that we shall never arrive at the indivisible, that is, the least common measure of the magnitudes? This then is matter of demonstration, whereas it is an axiom that everything continuous is divisible, so that a finite continuous line is divisible. The writer of the Elements bisects a finite straight line, starting from the latter notion, and not from any assumption that it is divisible without limit" Proclus adds that the proposition may also serve to refute Xenocrates' theory of indivisible lines (di-ojiu* ypapjiat). The argument given by Proclus to disprove the existence of indivisible lines is substantially that used by Aristotle as regards magnitudes generally (cf Physkt vi. 1, 231 a 21 sqq. and especially vl. 2, 133 b 15 — 32). J. n] PROPOSITIONS io, u 269 Proposition ii. To draw a straight line at right angles to a given straight line from a given point on it. Let AB be the given straight line, and C the given point on it. s Thus it is required to draw from the point C a straight line at right angles to the straight line AB. Let a point D be taken at ran- dom on AC; 10 let CE be made equal to CD ; [1. 3] on DE let the equilateral triangle FDE be constructed, [1. i] and let FC be joined ; I say that the straight line FC has been drawn at right is angles to the given straight line AB from C the given point on it. For, since DC is equal to CE, and CF is common, the two sides DC, CF are equal to the two sides EC, 20 CF respectively ; and the base DF is equal to the base FE ; therefore the angle DCF is equal to the angle ECF\ M] and they are adjacent angles. But, when a straight line set up on a straight line makes »s the adjacent angles equal to one another, each of the equal angles is right ; [Def. 10] therefore each of the angles DCF, FCE is right. Therefore the straight line CF has been drawn at right angles to the given straight line AB from the given point 30 C on it. Q. E. F. 10. let CB be made equal to CD. The verb is mbrSu which, as welt as the othet parts of Ktitiai, a constant iy used lor the passive of rWiftu " lo plats " ; and the latter word it constantly used in the sense of making, e.g., one straight line equal to another straight line. 1 )e Morgan remarks that this proposition, which is " to bisect the angle made by a straight line and its continuation " [i.e. a flat angle], should be a particular case of 1. 9, the constructions being the same. Thjs is certainly J7o BOOK I ['■ worth noting, though I doubt the advantage of rearranging the propositions in consequence. Apollonius gave a construction for this proposition (see P rod us, p. 282, 8) differing from Euclid's in much the same way as his construction for bisecting a straight line differed from that of 1. 10. Instead of assuming an equilateral triangle drawn without repeating the process of 1. 1, Apollonius takes D and E equidistant from C as in Euclid, and then draws circles in the manner of 1. 1 meeting at F This necessitates proving again that DF\s equal to FE\ whereas Euclid's assumption of the construction of 1. 1 in the words " let the equilateral triangle FDE be constructed " enables him to dispense with the drawing of circles and with the proof that DF is equal to FE at the same time. While however the substitution of Apollonius' constructions for 1. 10 and 1 1 would show faulty arrangement in a theoretical treatise like Euclid's, they are entirely suitable for what we call practical geometry, and such may have been Apollonius' object in these constructions and in his alternative for '- »3- . . . . Proclus gives a construction for drawing a straight line at right angles to another straight line but from one end of it, instead of from an intermediate point on it, it being supposed (for the sake of argument) that we are not permitted to produce the straight line. In the commentary of an-Nairizi (ed. Besthorn-Heiberg, pp. 73 — 4; ed. Curtze, pp. 54 — 5) this construction is attributed to Heron. Let it be required to draw from A a straight line at right angles to AB. On AB take any point C, and in the manner of the proposition draw CE at right angles to AB. from CE cut off CD equal to AC, bisect the angle ACE by the straight line CF\ [1. 9] and draw DF at right angles to CE meeting CF in F. Join FA. Then the angle FAC will be a right angle. For, since, in the triangles ACF, DCF, the two sides AC, CF are equal to the two sides DC, CF respectively, and the included angles ACF, DCFare equal, the triangles are equal in all respects. [1, 4] Therefore the angle at A is equal to the angle at D, and is accordingly a right angle. E F D A ( a PROPOSITION 12. To a given infinite straight tine, from a given point which is not on it, to draw a perpendicular straight line. Let AB be the given infinite straight line, and C the given point which is not on it ; I. u] PROPOSITIONS ii, 12 371 Sthus it is required to draw to the given infinite straight line AB, from the given point C which is not on it, a per- pendicular straight line. For let a point D be taken 10 at random on the other side of the straight line AB, and with centre C and distance CD let the circle EFG be described ; [Post. 3] let the straight line EG ij be bisected at H, [1. 10] and let the straight lines CG, CH, CE be joined. [Post. 1] I say that CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it. jo For, since GH is equal to HE, and HC is common, the two sides GH, HC are equal to the two sides EH, HC respectively ; and the base CG is equal to the base CE ; 35 therefore the angle CHG is equal to the angle EHC. [1. 8] And they are adjacent angles. But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is jo called a perpendicular to that on which it stands. [Def. 10] Therefore CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it Q. E. F. 1. a perpendicular straight line, tiitrer ti$ttv ypapph'- This is the full expression for a ptrpendkutar, KiBtrm meaning >tt f<il( or Ut down, so that the expression corresponds 10 our plumb-lint, if jcd&rar is however constantly used atone for a perpendicular, ypafi^ being understood. 10. on the other side of the straight line AB, literally " towards the other parts of the straight line AB," irl ri Irtpa ftfpif rt,s AB. Cf. "on the same side" (6rl ri au-ri pif"!) in Post, s *»d "in both directions" (ty inirtpa ri nipt)) in Def. 13. "This problem," says Proctus (p. 183, 7 — 10), "was first investigated by Oenopides [5 th cent b,c], who thought it useful for astronomy. He however calls the perpendicular, in the archaic manner, (a line drawn) 2j2 BOOK I [i. it gnomon-wise (koto yvu/iova), because the gnomon is also at right angles to the horizon." In this earlier sense the gnomon was a staff placed in a vertical position for the purpose of casting shadows and so serving as a means of measuring time (Cantor, Gezchichle der Mathemattk, i s , p. 161). The later meanings of the word as used in Eucl. Book n. and elsewhere will be explained in the note on Book n. Def. 2, Proclus says that two kinds of perpendicular were distinguished, the "plane" (iiriwtSos) and the "solid" (err«pea), the former being the perpendicular dropped on a line is a plane and the latter the perpendicular dropped on a plane. The term "solid perpendicular" is sufficiently curious, but it may perhaps be compared with the Greek term " solid locus " applied to a conic section, apparently on the ground that it has its origin in the section of a solid, namely a cone. Attention is called by most editors to the assumption in this proposition that, if only D be taken on the side of AB remote from C, the circle described with CD as radius must necessarily cut AB in two points. To satisfy us of this we need, as in I, 1, some postulate of continuity, e.g. something like that suggested by Killing (see note on the Principle of Continuity above, p. 235): "If a point [here the point describing the circle] moves in a figure which is divided into two parts [by the straight line J, and if it belongs at the beginning of the motion to one part and at another stage of the motion to the other part, it must during the motion cut the boundary between the two parts," and this of course applies to the motion in two directions from D. But the editors have not, as a rule, noticed a possible objection to the Euclidean statement of this problem which is much more difficult to dispose of at this stage, i.e. without employing any proposition later than this in Euclid's order. How do we know, says the supposed critic, that the circle does not cut AB in three or more points, in which case there would be not one perpendicular but three or more? Proclus (pp. 186, 12—489, *>) tries to refute this objection, and it is interesting to follow his argument, though it will easily be seen to be inconclusive. He takes in order three possible suppositions. 1. May not the circle meet AB in a third point K between the middle point of GE and either extremity of it, taking the form drawn in the figure appended ? Suppose this possible. Bisect GE in H. Join CH, and produce it to meet the circle in L. Join CG, CK, CE. Then, since CG is equal to CE, and CH is common, while the base GH is equal to the base HE, the angles CHG, CHE are equal and, since they are adjacent, they are both right. Again, since CG is equal to CE, the angles at G and E are equal. Lastly, since CK is equal to CG and also to CE, the angles CGK, CKG are equal, as also are the angles CKE, CEK. Since the angles CGK, CEK are equal, it follows that the angles CKG, CKE are equal and therefore both right. Therefore the angle CKH'\% equal to the angle CHK, and CH is equal to CK. I. 12] PROPOSITION 12 273 But CK is equal to CL, by the definition of the circle ; therefore CH is equal to CL : which is impossible. Thus Proclus; but why should not the circle meet AB in H as well as .X? 2. May not the circle meet AB in // the middle point of GE and take the form shown in the second figure? In that case, says Proclus, join CG, CH, CE as before. Then bisect ME at K, join CK and produce it to meet the circumference at L. Now, since HK is equal to KE, CK is common, and the base CH is equal to the base CE, the angles at K are equal and therefore both right angles. Therefore the angle CHK is equal to the angle CKH, whence CK is equal to CH and therefore to CL : which is impossible. So Proclus ; but why should not the circle meet AB in A* as well as Hf 3, May not the circle meet AB in two points besides G, E and pass, between those two points, to the side of A3 towards C, as in the next figure ? Here again, by the same method, Proclus proves that, K, L being the other two points in which the circle cuts AB, CK is equal to CH, and, since the circle cuts CH'm M, CM is equal to CK and therefore to CH: which is impossible. But, again, why should the circle not cut AB in the point H a& well? In fact, Proclus' cases are not mutually exclusive, and his method of proof only enables us to show that, if the circle meets AB in one more point besides G, E, it must meet it in more points still. We can always find a new point of intersection by bisecting the distance separating any two points of intersection, and so, applying the method ad infinitum, we should have to conclude ultimately that the circle with radius CH (or CG) coincides with AB. It would follow that a circle with centre C and radius greater than CH would not meet AB at all. Also, since all straight lines from C to points on AB would be equal in length, there would be an infinite number of perpendiculars from C on AB. Is this under any circumstances possible ? It is not possible in Euclidean space, but it is possible, under the Riemann hypothesis (where a straight line is a " closed series " and returns on itself), in the case where C is the pole of the straight line AB. It is natural therefore that, for a proof that in Euclidean space there is only one perpendicular from a point to a straight line, we have to wait until 1. 16, the precise proposition which under the Riemann hypothesis is only valid with a certain restriction and not universally. There is no difficulty involved by waiting until 1. 16, since t. 12 is not used before that proposition is reached; and we are only in the same position as when, in order to satisfy ourselves of the number of possible solutions of 1. r, we have to wait till 1. 7. But if we wish, after all, to prove the truth of the assumption without recourse to any later proposition than 1. 12, we can do so by means of this same invaluable 1. 7. »74 BOOK I [l, 12 If the circle intersects AB as before in G, E, let H be the middle point of GE, and suppose, if possible, that the circle also intersects AB in any other point K on AH. From H, on the side of AB opposite to C, draw HL at right angles to AB, and make HL equal to HC. Join CG, EG, CK, LK. Now, in the triangles CHG, EHG, CHis equal to /.//, and HG is common. Also the angles CHG, EHG, being both right, are equal. Therefore the base CG is equal to the base EG. Similarly we prove that CK is equal to LK. But, by hypothesis, since K is on the circle, CK is equal to CG. Therefore CG, CK, LG, LK are all equal. Now the next proposition, i. 13, will tell us that CH, HL are in a straight line; but we will not assume this. Join CE, Then on the same base CE and on the same side of it we have two pairs of straight lines drawn from C, L to G and K such that CG is equal to CK and EG to LK But this is impossible [1. 7 \ Therefore the circle cannot cut BA or BA produced in any point other than G on that side of CL on which G is. Similarly it cannot cut AB or AB produced at any point other than E on the other side of CL. The only possibility le/t therefore is that the circle might cut AB in the same point as that in which CL cuts it. But this is shown to be impossible by an adaptation of the proof of I. 7, For the assumption is that there may be some point M on CL such thai CM is equal to CG and LM to LG. If possible, let this be the case, and produce CG to N. Then, since CM is equal to CG, the angle NGM is equal to the angle GML [1. 5, part 2]. Therefore the angle GML is greater than the angle MGL. Again, since LG k equal to LM, the angle GML is equal to the angle MGL. But it was also greater : which is impossible. Hence the circle in the original figure cannot cut AB in the point in which CL cuts it. Therefore the circle cannot cut AB in any point whatever except G and E. [This proof of course does not prove that CK is less than CG, but only that it is not equal to it. The proposition that, of the obliques drawn from C to AB, that is less the foot of which is nearer to H can only be proved later. The proof by 1. 7 also fails, under the Riemann hypothesis, if C, L are the poles of the straight line AB, since the broken lines CGL, CKL etc. become equal straight lines, all perpendicular to AB.] Proclus rightly adds (p. z8q, 18 sqq.) that it is not mcessary to take D on the side of AB away from A if an objector " says that there is no space on 1. is, 13] PROPOSITIONS is, 13 »75 that side." If it is not desired to trespass on that side of AB, we can take D anywhere on AB and describe the arc of a circle between D and the point where it meets AB again, drawing the arc on the side of AB on which C is. If it should happen that the selected point D is such that the circle only meets AB in oik point (D itself), we have only to describe the circle with CD as radius, then, if E be a point on this circle, take Fa point further from C than E is, and describe with CF as radius the circular arc meeting AB in two points. Proposition 13. If a straight line set up on a straight line make angles, it wilt make either two right angles or angles equal to two right angles. For let any straight line AB set up on the straight line s CD make the angles CBA, ABD j I say that the angles CBA, ABD are either two right angles or equal to two right angles. Now, if the angle CBA is equal to 10 the angle ABD, they are two right angles. [Def. 10] But, if not, let BE be drawn from the point B at right angles to CD ; [1. n] therefore the angles CBE, EBD are two right angles. is Then, since the angle CBE is equal to the two angles CBA, ABE, let the angle EBD be added to each ; therefore the angles CBE, EBD are equal to the three angles CBA, ABE, EBD. [C X a] M Again, since the angle DBA is equal to the two angles DBE, EBA, let the angle ABC be added to each ; therefore the angles DBA. ABC are equal to the three angles DBE, EBA, ABC. [ax*] ij But the angles CBE, EBD were also proved equal to the same three angles ; and things which are equal to the same thing are also equal to one another ; [C. X 1] therefore the angles CBE, EBD are also equal to the 3a angles DBA, ABC. 376 BOOK I [i. 13, 14 But the angles CBE, EBD are two right angles ; therefore the angles DBA, ABC are also equal to two right angles. Therefore etc. Q. E, D, 17. let the angle EBD be added to each, literally "let the angle EBD be added (so as to be) common, " *a*fy rpoandeBu % irrh EBA. Similarly tow)) d^pjsrtfw is used of subtracting a straight line or angle from each of two others. "Let the common angle EBD be added is clearly an inaccurate translation, for the angle is not common before it 13 added, i.e. the mur^i is proleptie. "Let the common angle be tuitraclid" as a translation of rar^ d<t?W?l<j8u would be less unsatisfactory, it is true, but, as it is desirable to use corresponding words when translating the two expressions, it seems hopeless to attempt to keep the word "common,'' and I have therefore said " to each" and "from each. " simply. Proposition 14. If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another. 5 For with any straight line AB, and at the point B on it, let the two straight lines BC, BD not lying on the same side make the adjacent angles ABC, ABD equal to two right angles ; I say that BD is in a straight line with CB. w For, if BD is not in a straight line with BC, let BE be in a straight line with CB. Then, since the straight line AB 5 — stands on the straight line CBE, 15 the angles ABC, ABE are equal to two right angles. [<■ n] But the angles ABC, ABD are also equal to two right angles ; therefore the angles CBA, ABE are equal to the angles CBA, ABD. [Post 4 and C. N. t] Let the angle CBA be subtracted from each ; 20 therefore the remaining angle ABE is equal to the remaining angle ABD, [C. N. 3] the less to the greater : which is impossible. Therefore BE is not in a straight line with CB. Similarly we can prove that neither is any other straight aj line except BD. I. 14. IS] PROPOSITIONS 13—15 277 Therefore CB is in a straight line with BD. Therefore etc. Q. E. D. i. If with any straight line... There is no greater difficulty in translating the works of the Greek geometers than that of accurately giving the force of prepositions, *pbi, for instance, is used in all sorts of expressions with various shades of meaning. The present enunciation begins 'EAr Tp&t Tttfi ciffcta rai tQ rpAi avrv eTtfttitp, and it is really necessary in this one sentence to translate rp6i by three different words, wit A, at, and en. The first rp6t must be translated by with because two straight lines " make" an angle with one another. On the other hand, where [he similar expression rpot rp bofalfy t&dcta occurs 1 in I. 33, but it is a question of M constructing ™ an angle (riPcrr^ffaffPat), we have to say "to construct on a given straight line." Againtt would perhaps be the English word coming nearest to expressing all these meanings of irp£t, but it would be intolerable as a translation. 17. Todhunter points out tbat for the inference in this line Post. 4, that all right angles are equal, is necessary as well as the Common Notion that things which are equal to the same thing [or rather, here, to squat things) are equal. A similar remark applies to steps in the proofs of I. 15 and I. 18. 34. we can prove. The Greek expresses this by the future of the verb, &*t%o/ur, "we shall prove," which however would perhaps be misleading in English. P rocki 5 observes (p. 297) that two straight lines on the same side of another straight line and meeting it in one and the same point may make with one and the same portion of the straight line terminated at the point two angles which are together equal to two right angles, in which case however the two straight lines would not be in a straight line with one another. And he quotes from Porphyry a construction for two such straight lines in the particular case where they form with the given straight line angles equal respectively to half a right angle and one and a half right angles. There is no particular value in the construction, which will be gathered from the annexed figure where CB, CF are drawn at the prescribed inclinations to CD. Proposition 15. If two straight lines cut one another, they make the vertical angles equal to one another. For let the straight lines AB, CD cut one another at the point E ; 5 I say that the angle AEC is equal to i^^ the angle DEB, _\I_ and the angle CEB to the angle D AED. For, since the straight line AE stands 10 on the straight line CD, making the angles CEA, AED, the angles CEA, AED are equal to two right angles L'- '3] »78 BOOK I [l tS Again, since the straight line DE stands on the straight line AB, making the angles AED, DEB, the angles AED, DEB are equal to two right angles. [«■ '3] «s But the angles CEA, A ED were also proved equal to two right angles ; therefore the angles CEA, A ED are equal to the angles AED DEE. [Post 4 and C. If. 1] Let the angle AED be subtracted from each ; *> therefore the remaining angle CEA is equal to the remaining angle BED. [C. If. 3] Similarly it can be proved that the angles CEB, DEA are also equal. Therefore etc. Q. E, D. *5 [Porism. From this it is manifest that, if two straight lines cut one another, they will make the angles at the point of section equal to four right angles.] 1. the vertical angles. The difference between adjacent angles (ai ifeffit fut/ku) and vertical fugles (td Kara xopv<pfyv yuvttu) is t hi lis explained by Proclus (p. 398, 14—34). The first term describes the angles made by two straight lines when one only it divided by the other, i.e. when one straight Line meets another at a point which is not either of its extremi- ties, but is not itself produced beyond the point of meeting. When the first straight line is produced, so that the lines cross at the point, they make two pairs of vertical angles (which are more clearly described as vertically opposite angles), and which are so called because their convergence is from opposite directions to one point (the intersection of the lines) as vertex 16. at the point of section, literally " at the section," rpii tjj 7-0^5, This theorem, according to Eudemus, was first discovered by Thales, but found its scientific demonstration in Euclid (Proc)us, p. 2991, 3 — 6). Proclus gives a converse theorem which may be stated thus. If a straight line is met at one and the same point intermediate in its length by two other straight lines on different sides of it and such as to make the vertical angles equal, the latter straight lines are in a straight line with one another. The proof need not be given, since it is almost self-evident, whether (1) it is direct, by means of 1. 13, 14, or (2) indirect, by reductio ad abmrdum depending on 1. 15. The balance of ms. authority seems to be against the genuineness of this Porism, but Proclus and Psellus both have it. The word is not here used, as it is in the title of Euclid's lost Potisms, to signify a particular class of independent propositions which Proclus describes as being in some sort intermediate between theorems and problems (requiring us, not to bring a thing into existence, but to find something which we know to exist). Porism has here (and wherever the term is used in the Elements) its second meaning ; it is what we call a corollary, i.e. an incidental result springing from the proof of a theorem or the solution of a problem, a result not directly sought but appearing as it were by chance without any additional labour, and constituting, as Proclus says, a sort of windfall (ippaiov) and bonus (ntpSos). These Porisms appear in both the i. 15, i6] PROPOSITIONS 15, 16 179 geometrical and arithmetical Books of the Elements, and may either result from theorems or problems. Here the Porism is geometrical, and springs out of a theorem ; vit. 1 affords an instance of an arithmetical Porism. As an instance of a Porism to a problem Proclus cites " that which is found in the second Book" (ri b t^ Stvtipif jSt^Aup mi^mfov); but as to this see notes on 11. 4 and iv. 15. The present Porism, says Proclus, formed the basis of " that paradoxical theorem which proves that only the following three (regular) polygons can fill up the whole space surrounding one point, the equilateral triangle, the square, and the equilateral and equiangular hexagon." We can in fact place round a point in this manner six equilateral triangles, three regular hexagons, or four squares. "But only the angles of these regular figures, to the number specified, can make up four right angles : a theorem due to the Pythagoreans." Proclus further adds that it results from the Porism that, if any number of straight lines intersect one another at one point, the sum of all the angles so formed will still be equal to four right angles. This is of course what is generally given in the text-boots as Corollary 1. Proposition 16, In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles. Let ABC be a triangle, and let one side of it BC be produced to D ; l I say that the exterior angle ACD is greater than either of the interior and opposite angles CBA, BAC. Let AC be bisected at E [t 10], and let BE be joined and produced 10 in a straight line to F ; let EFbe made equal to BE[t. 3], let EC be joined [Post. i],and let AC be drawn through to G [Post. a]. Then, since AE is equal to EC, 15 and BE to EF, the two sides AE, EB are equal to the two sides CE, EF respectively ; and the angle AEB is equal to the angle FEC, for they are vertical angles. [1. 15] » Therefore the base AB is equal to the base FC, and the triangle ABE is equal to the triangle CFE, and the remaining angles are equal to the remaining angles respectively, namely those which the equal sides subtend ; [1. 4] therefore the angle BAE is equal to the angle ECF. i8o BOOK I [i. 16 i But the angle BCD is greater than the angle ECF; [C.N. 5] therefore the angle A CD is greater than the angle BAB. Similarly also, if BC be bisected, the angle BCG, that is, the angle ACD [i. is], can be proved greater than the angle ABC as well. Therefore etc. Q. E. D. i. the exterior angle, literally "the outside angle," <\ l«rdt yurla. i. the interior and opposite angles, rut inrit Kai iritarrior yuri&r. 11. let AC be drawn through to G. The word is 5iiJx* u p a variation on Ihe more usual 4K$ffi\-fiadw. "let it be praduHd. " at. CFE, in the text " F£C," As is well known, this proposition is not universally true under the Riemann hypothesis of a space endless in extent but not infinite in size. On this hypothesis a straight line is a "closed series" and returns on itself; and two straight lines which have one point of intersection have another point of intersection also, which bisects the whole length of the straight line measured from the first point on it to the same point again; thus the axiom of Euclidean geometry that two straight lines do not enclose a space does not hold. If 4 A denotes the finite length of a straight line measured from any point once round to the same point again, 2A is the distance between the two intersections of two straight lines which meet. Two points A, B do not determine one sole straight line unless the distance between them is different from 2 A. In order that there may only be one perpendicular from a point C to a straight line AS, C must not be one of the two " poles " of the straight line. Now, in order that the proof of the present proposition may be universally valid, it is necessary that CF should always fall within the angle ACD so that, the angle A CF may be less than the angle ACD. But this will not always be so on the Riemann hypothesis. For, (1) if BE is equal to A, so that BF is equal to 2 A, .Fwill be the second point in which BE and BD intersect ; Le. F will lie on CD, and the angle ACF will be equal to the angle ACD. In this case the exterior angle ACD will be equal to the interior angle BAC. (2) If BE is greater than A and less than 2 A, so that BF is greater than a A and less than 4A, the angle ACF will be greater than the angle ACD, and therefore the angle A CD will be less than the interior angle BAC. Thus, e.g., in the particular case of a right-angled triangle, the angles other than the right angle may be (1) both acute, (2) one acute and one obtuse, or (3) both obtuse according as the perpendicular sides are (1) both less than A, (2) one less and the other greater than A, (3) both greater than A. Proclus tells us (p. 307, 1 — 12) that some combined this theorem with the next in one enunciation thus : In any triangle, if one side be produced, the exterior angle of the triangle is greater than either of the inierior and opposite angles, and any two of the interior angles are less than two right angles, the combination having been suggested by the similar enunciation of Euclid 1. 32, In any triangle, if one of the suits be produced, the exterior angle is equal to the two inierior and opposite angles, and the three interior angles of the triangle are equal to two right angles. The present proposition enables Proclus to prove what he did not succeed in establishing conclusively in his note on 1. 12, namely that from one point- there cannot be drawn to the same straight line three straight lines equal in length. i6, 17I PROPOSITIONS 16, 17 tii For, if possible, let AB, AC, AD be all equal, B, C, D being in a straight line. Then, since AB, AC are equal, the angles ABC, ACB are equal. Similarly, since AB, AD are equal, the angles ABD, ADB are equal. Therefore the angle ACB is equal to the angle ADC, i.e. the exterior angle to the interior and opposite angle : which is impossible. Proclus next {p. 308, 14 sqq.) undertakes to prove by means of 1. 16 that, if a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle, the two straight lines will not form a triangle or meet, for in that case the same angle would be both greater and equal. The proof is really equivalent to that of Eucl. 1. 17, If BE falls on the two straight lines AB, CD in such a way that the angle CDE is equal to the interior and opposite angle ABD, AB and CD cannot form a triangle or meet. For, if they did, then (by 1. 16) the angle CDE would be greater than the angle ABD, while by the hypothesis it is at the same time equal to it. Hence, says Proclus, in order that BA, DC may form a triangle it is necessary for them to approaeh one another in the sense of being turned round one pair of corresponding extremities, e.g. B, D, so that the other extremities A, C come nearer. This may be brought about in one of three ways: (t) AB may remain fixed and CD be turned about D so that the angle CDE increases ; (2) CD may remain fixed and AB be turned about B so that the angle ABD b?r jmes smaller ; (3) both AB and CD may move so as to make the angle ABD smaller and the angle CDE larger at the same time. The reason, then, of the straight lines AB, CD coming to form a triangle or to meet is (says Proclus) the movement of the straight lines. Though he does not mention it here, Proclus does in another passage (p. 371, 2 — ro, quoted on p. 207 above) hint at the possibility that, while 1. 16 may remain universally true, either of the straight lines BA, DC (or both together) may be turned through any angle not greater than a certain finite angle and yet may not meet (the Bolyai-Lobachewsky hypothesis). Proposition 17. In any triangle two angles taken together in any manner are less than two right angles. Let ABC be a triangle ; I say that two angles of the triangle ABC taken together in any manner are less than two right angles. For let BC be produced to D. [Post. 2] Then, since the angle ACD is an exterior angle of the triangle ABC, it is greater than the interior and opposite angle ABC. 282 BOOK I [l 17 Let the angle ACB be added to each ; therefore the angles A CD, ACB are greater than the angles ABC, BCA. But the angles A CD, ACB are equal to two right angles. ['; '3] Therefore the angles ABC, BCA are less than two right angles. Similarly we can prove that the angles BAC, ACB are also less than two right angles, and so are the angles CAB, ABC as well. Therefore etc. Q. E. D. 1. taken together in any manner, rirrg pcraXop0artyui'eu, Le. an; pail added together- As in his note on the previous proposition, Prod us tries to state the cause of the property. He takes the case of two straight lines forming right angles with a transversal and observes that it is the convergence of the straight lines towards one another (<n!™«rn t&v tvBuuf), the lessening of the two right angles, which produces the triangle. He will not have it that the fact of the exterior angle being greater than the interior and opposite angle is the cause of the property, for the odd reason that "it is not necessary that a side should be produced, or that there should be any exterior angle constructed. ..and how can what is not necessary be the cause of what is necessary ?" (p. 311, 17 — 21). Agreeably to this view, Proclus then sets himself to prove the theorem without producing a side of the triangle. Let ABC be a triangle. Take any point D on BC, and join AD. Then the exterior angle A DC of the triangle ABD is greater than the interior and opposite angle ABD. Similarly the exterior angle ADB of the triangle ADC is greater than the interior and opposite angle A CD. Therefore, by addition, the angles ADB, ADC are together greater than the angles ABC, ACB. But the angles ADB, ADC are equal to two right angles ; therefore the angles ABC, ACB are less than two right angles. Lastly, Proclus proves (what is obvious from this proposition) that there cannot be more than one perpendicular to a straight line from a point without it. For, if this were possible, two of such perpendiculars would form a triangle in which two angles would be right angles: which is impossible, since any two angles of a triangle are together less than two right angles. i. 18] PROPOSITIONS 17, 18 J83 Proposition 18. In any triangle the greater side subtends the greater angle. For let ABC be a triangle having the side AC greater than AB ; I say that the angle ABC is also greater than the angle BCA. For, since AC is greater than AB, let AD be made equal to AB [i. 3], and let BD be joined. Then, since the angle ADB is an exterior angle of the triangle BCD, it is greater than the interior and opposite angle DCB. [1. 16] But the angle ADB is equal to the angle ABD, since the side AB is equal to AD ; therefore the angle ABD is also greater than the angle ACB\ therefore the angle ABC is much greater than the angle ACB. Therefore etc. Q. E. D. In the enunciation of this proposition wc have inrarttvay (*" subtend "} used with the simple accusative instead of the more usual inti with accusative. The latter construction is used in the enunciation of I. ig» which otherwise only diners from that of [• 18 in the order of the words. The point to remember in order to distinguish the two is that the datum comes first and the fuanititm second, the datum being in this proposition the greater sidt and in the next the greater angle. Thns the enunciations are (i. 18) i-a»n6i i-ftydtov j) pclpgr J-XrupA tjj* fitlfrtrvL ywflsw uwordvti and (l. 19) rarrbt rpvy&wou vxb rV pd{ot>a -yuvLar Jj licifuw xXcupi. iinoTtltti. In order to keep the proper order in English we must use the passive of the verb in I. 19. Aristotle quotes the result of 1. to, using the exact wording, vk& fkp t)jv ^ttflfw ywrtay bwoTttt/ci {Mtteorologica tit. 5, 376 a 11). " In order to assist the student in remembering which of these two propositions [1. 18, 19] is demonstrated directly and which indirectly, it may be observed that the order is similar to that in I. 5 and 1. 6" (Todhunter). An alternative proof of L 18 given by Porpnyry (see Ptoclus, pp. 315, \\ — 316, 13} is interesting. It starts by supposing a length equal to AB cut off from the other end of AC; that is, CD and not AD is made equal to AB. Produce AB to E so that BE is equal to AD, and join EC. Then, since AB is equal to CD, and BE to AD, AE is equal to AC, aS* BOOK I [i. 18, 19 Therefore the angle A EC is equal to the angle ACE. Now the angle ABC is greater than the angle A EC, [t. 16] and therefore greater than the angle ACE. Hence, a fortiori, the angle ABC is greater than the angle ACB. Proposition 19. In any triangle the greater angle is subtended by the greater side. Let ABC be a triangle having the angle ABC greater than the angle BCA ; I say that the side AC is also greater than the side AB. For, if not, AC is either equal to AB or less. Now AC is not equal to AB ; for then the angle ABC would also have been equal to the angle A CB ; [u 5] but it is not ; therefore AC is not equal to AB. Neither is AC less than AB, for then the angle ABC would also have been less than the angle ACB; [t 18] but it is not ; therefore AC is not less than AB. And it was proved that it is not equal either. Therefore AC is greater than AB, Therefore etc. q. e. d. This proposition, like t. 6, can be proved by merely logical deduction from 1. 5 and 1. 18 taken together, as pointed out by De Morgan. The general form of the argument used by De Morgan is given in his Formal Logic (1847), p. 25, thus : "Hypothesis. Let there be any number of propositions or assertions — three for instance, X, Y and Z — of which it is the property that one or the other must be true, and one only. Let there be three other propositions P, Q and P of which it is also the property that one, and one only, must be true. Let it be a connexion of those assertions that : when X is true, P is true, when Kis true, Q is true, when Z is true, R is true. Constquenu : then it follows that, when P is true, X is true, when Q is true, Y is true, when JR is true, Z is true" [1 ■5] [<• !8] [' .6] [«■ , 9 ] i. 19] PROPOSITIONS 18, 19 185 To apply this to the case before us, let us denote the sides of the triangle ABC by a, 6, c, and the angles opposite to these sides by A, B, C respectively, and suppose that a is the base. Then we have the three propositions, when b is equal to c, B is equal to C, when b is greater than c, B is greater than C, 1 when 6 is less than c, B is less than C, f and it follows logically that, when B is equal to C, b is equal to c, when B is greater than C, b is greater than c, \ when i? is less than C, b is less than c. J Reductio ad absurdum by exhaustion. Here, says Proclus (p. 318, 16 — 33), Euclid proves the impossibility "by means of division" (« fkaipurtan). This means simply the separation of different hypotheses, each of which is inconsistent with the truth of the theorem to be proved, and which therefore must be successively shown to be impossible. If a straight line is not greater than a straight tine, it must be either equal to it or less ; thus in a reductio ad absurdum intended to prove such a theorem as 1. 19 it is necessary to dispose successively of two hypotheses inconsistent with the truth of the theorem. Alternative (direct) proof. Proclus gives a direct proof (pp. 319—321) which an-NairizI also has and attributes to Heron. It requires a lemma and is consequently open to the slight objection of separating a theorem from its converse. But the lemma and proof are worth giving. Lemma. If an angle of a triangle be bisected and the straight line bisecting it meet the base and divide it into unequal parts, the sides containing the angle will be unequal, and the greater will be that which meets the greater segment of the base, and the less that which meets the lest. Let AD, the bisector of the angle A of the triangle ABC, meet BC in D, making CD greater than BD. I say that AC is greater than AB. Produce AD to £ so that DE is equal to AD. And, since DC is greater than BD, cut off DF equal to BD. Join BFsmd produce it to G. Then, since the two sides AD, DB are equal to the two sides ED, DF, and the vertical angles at D are equal, AB is equal to EF, and the angle DEF to the angle BAD, i.e. to the angle DAG (by hypothesis). Therefore AG is equal to EG, and therefore greater than EF, or AB, Hence, a fortiori, AC is greater than AB. *86 BOOK I U. 19, 30 Proof of I. 19. Let ABC be a triangle in which the angle ABC is greater than the angle ACB. * Bisect BC at D, join AD, and produce it to B so that DE is equal to ^Z>. Join BE. Then the iwo sides BD t DE are equal to the two sides CD, DA, and the vertical angles at D are equal ; therefore BE is equal to AC, and the angle DBE to the angle at C. But the angle at C is less than the angle ABC ; therefore the angle DBE is less than the angle ABD. Hence, if BF bisect the angle ABE, BF meets AE between A and D. Therefore EF is greater than FA. It follows, by the lemma, that BE is greater than BA, that is, AC is greater than >4.#. Proposition 20. /« a«y triangle twu sides taken together in any manner are greater than the remaining one. For let ABC be a triangle ; I say that in the triangle ABC two sides taken together in any manner are greater than the remaining one, namely BA, AC greater than BC, AB, BC greater than A C, BC, CA greater than AB. For let BA be drawn through to the point D s let DA be made equal to CA, and let DC be joined. Then, since DA is equal to AC, the angle ADC is also equal to the angle ACD; [1. 5 ] therefore the angle BCD is greater than the angle ADC. [C. JV. 5] And, since DCB is a triangle having the angle BCD greater than the angle BDC, and the greater angle is subtended by the greater side, [l 19] therefore DB is greater than BC. i. 20] PROPOSITIONS 19, 20 287 ButZM is equal to AC; therefore BA, AC are greater than BC. Similarly we can prove that AB, BC are also greater than CA, and BC, CA than AB. Therefore etc. Q. E. D, It was the habit of the Epicureans, says Prod us (p. 322), to ridicule this theorem as being evident even to an ass and requiring no proof, and their allegation that the theorem was "known" (yywpijiov) even to an ass was based on the fact that, if fodder is placed at one angular point and the ass at another, he does not, in order to get to his food, traverse the two sides of the triangle but only the one side separating them (an argument which makes Savile exclaim that its authors were "digni ipsi, qui cum Asino foenum essent," p. 78). Proclus replies truly that a mere perception of the truth of the theorem is a different thing from a scientific proof of it and a knowledge of the reason why it is true. Moreover, as Sim son says, the number of axioms should not be increased without necessity. Alternative Proofs. Heron and Porphyry, we are told (Proclus, pp. 323 — 6), proved this theorem in different ways as follows, without producing one of the sides. First proof. Let ABC be the triangle, and let it be required to prove that the sides BA, AC are greater than BC. Bisect the angle BAC by AD meeting BC inD. Then, in the triangle ABD, the exterior angle ADC is greater than the interior and opposite angle BAD, [1. 16] that is, greater than the angle DAC. Therefore the side AC is greater than the side CD, [1. 19] Similarly we can prove that AB is greater than BD. Hence, by addition, BA, AC are greater than BC. Second proof. This, like the first proof, is direct. There are several cases to be considered. (1) If the triangle is equilateral, the truth of the proposition is obvious. (3) If the triangle is isosceles, the proposition needs no proof in the case (a) where each of the equal sides is greater than the base. (#) If the base is greater than either of the other sides, we have to prove that the sum of the two equal sides is greater than the base. Let BC be the base in such a triangle. Cut off from BC a length BD equal to AB, and join AD. Then, in the triangle ADB, the exterior angle ADC is greater than the interior and opposite angle BAD. [1. 1 61 Similarly, in the triangle ADC, the exterior angle ADB is greater than the interior and opposite angle CAD. *88 BOOK I [i. to By addition, the tyro angles BDA, ADC are together greater than the two angles BA D, DA C (or the whole angle BA C). Subtracting the equal angles BDA, BAD, we have the angle ADC greater than the angle CAD. It follows that AC is greater than CD; [i. 19] and, adding the equals AB, BD respectively, we have BA, AC together greater than BC. (3) If the triangle be scalene, we can arrange the sides in order of length. Suppose BC is the greatest, AB the intermediate and AC the least side. Then it is obvious that AB, BC are together greater than AC, and BC, CA together greater than AB. It only remains therefore to prove that CA, AB are together greater than BC. We cut off from BC a length BD equal to the adjacent side, join AD, and proceed exactly as in the above case of the isosceles triangle. Thirdprwf. This proof is by reduetio ad ahsurdum. Suppose that BC is the greatest side and, as before, we have to prove that BA, AC are greater than BC. If they are not, they must be either equal to A or less than BC. (1) Suppose BA, AC ire together equal to BC. From BC cut off BD equal to BA, and join AD. It follows from the hypothesis that DC is equal to AC. Then, since BA is equal to BD, the angle BDA is equal to the angle BAD. Similarly, since AC is equal to CD, the angle CDA is equal to the angle CAD. By addition, the angles BDA, ADC are together equal to the whole angle BAC. That is, the angle BAC is equal to two right angles : which is impossible. {2) Suppose BA, AC ate together less than BC. From BC cut off BD equal to BA, and from CB cut off CE equal to CA. Join AD, AE. In this case, we prove in the same way that the angle BDA is equal to the angle BAD, and the angle CEA to the angle CAE. By addition, the sum of the angles BDA, AEC is equal to the sum of the angles BAD, CAE. Now, by 1. 16, the angle BDA is greater than the angle DAC, and therefore, a fortiori, greater than the angle EA C. Similarly the angle AEC is greater than the angle BAD. Hence the sum of the angles BDA, AEC is greater than the sum of the angles BAD, EAC. But the former sum was also equal to the latter : which is impossible, li ai] PROPOSITIONS 20, 21 289 Proposition 21. If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater s angle. On BC, one of the sides of the triangle ABC, from its extremities B, C, let the two straight lines BD, DC be con- structed meeting within the triangle ; I say that BD, DC are less than the remaining two sides 10 of the triangle BA, AC, but contain an angle BDC greater than the angle BAC. For let BD be drawn through to E. Then, since in any triangle two sides are greater than the remaining 15 one, [1. zo] therefore, in the triangle ABE, the two sides A B, AE are greater than BE. Let EC be added to each ; therefore BA, AC art greater than BE, EC. 20 Again, since, in the triangle CED, the two sides CE, ED are greater than CD, let DB be added to each ; therefore CE, EB are greater than CD, DB. But BA, AC were proved greater than BE, EC; 2$ therefore BA, AC are much greater than BD, DC. Again, since in any triangle the exterior angle is greater than the interior and opposite angle, [1- 16] therefore, in the triangle CDE, the exterior angle BDC is greater than the angle CED. 30 For the same reason, moreover, in the triangle ABE also, the exterior angle CEB is greater than the angle BAC. But the angle BDC was proved greater than the angle CEB ; therefore the angle BDC is much greater than the angle BAC. 3 5 Therefore etc. q. e. d, 1* be con strutted. ..meeting within the triangle. The word n meeting" is not in the Greek, where the words are irrht rwmrt&rty. evrLrrwiBtu is the word used of con- structing two straight lines to a point (cf- [■ 7) or so as to form a triangle ; but it is necessary in English to indicate that they mat. 3. the straight lines so constructed. Observe the elegant brevity of the Creek al 29° BOOK I [i. at The editors generally call attention to the fact that the lines drawn within the triangle in this proposition must be drawn, as the enunciation says, from the ends of the side ; otherwise it is not necessary that their sum should be less than that of the remaining sides of the triangle. Proclus (p. 3*7, II sqq.) gives a simple illustration. Let ABC be a right-angled triangle. Take any point D on BC, join DA, and cut off from it DE equal to AB. Bisect AE at F, and join FC. Then shall CF, FD be together greater than CA, AB. For CF, FE are equal to CF, FA, and therefore greater than CA. Add the equals ED, AB respectively ; therefore CF, FD are together greater than CA, AB. Pappus gives the same proposition as that just proved, but follows it up by a number of others more elaborate in character, selected apparently from " the so-called paradoxes " of one Erycinus (Pappus, m. p. 106 sqq.). Thus he proves the following : 1. In any triangle, except an equilateral triangle or an isosceles triangle with base less than one of the other sides, it is possible to construct on the base and within the triangle two straight lines the sum of which is equal to the sum of the other two sides of the triangle. 2. In any triangle in which it is possible to construct two straight lines on the base which are equal to the sum of the other two sides of the triangle it is also possible to construct two others the sum of which is greater than that sum. 3. Under the same conditions, if the base is greater than either of the other two sides, two straight tines can be constructed in the manner described which are respectively greater than the other two sides of the triangle ; and the lines may be constructed so as to be respectively equal to the two sides, if one of those two sides is less than the other and each of them less than the base. 4. The lines may be so constructed that their sum will bear to the sum of the two sides of the triangle any ratio less than 2:1. As a specimen of the proofs we will give that of the proposition which has been numbered (1) for the case where the triangle is isosceles (Pappus, in. pp. 108 — 110)1 I. »i] PROPOSITION 21 391 Let ABC be an isosceles triangle in which the base AC is greater than either of the equal sides .<4.5, BC. With centre v4 and radius AB describe a circle meeting j4Cin D. Draw any radius AEFsuch that it meets BC in a point F outside the circle. Take any point G on EF, and through it draw GZf parallel to AC. Take any point .AT on GH, and draw KL parallel to FA meeting AC in L. From jSCcut off BN equal to EG. Thus AG, or LK, is equal to the sum of AB, BN, and CWis less than LK. Now GF, Fffare together greater than GH, and CH, UK together greater than CK, Therefore, by addition, CF, FG, HK are together greater than CK, HG. Subtracting HK from each side, we see that CF, FG are together greater than CK, KG ; therefore, if we add AG to each, AF, FCaie together greater than AG, GK, KC. And AB, BC are together greater than AF, EC. [1. 31] Therefore AB, BC are together greater than A G, GK, KC. But, by construction, AB, BN are together equal to AG ; therefore, by subtraction, NC is greater than GK, KC, and a fortiori greater than KC. Take on KC produced a point .A/' such that KM is equal to NC; with centre K and radius KM describe a circle meeting CL in 0, and join KO. Then shall LK, KO be equal to AB, BC. For, by construction, LK is equal to the sum of AB, BN, and KO is equal to NC; therefore LK, KO are together equal to AB, BC. It is after 1. at that (as remarked by De Morgan) the important proposition about the perpendicular and obliques drawn from a point to a straight line of unlimited length is best introduced : Of all straight lines that can be drawn to a given straight line of unlimited length from a given point without it : (a) the perpendicular is the shortest ; (b) of the obliques, that is the greater the fool of which is further from the perpendicular ; (e) given one oblique, only one other can be found of the same length, namely that the foot of which is equally distant with the foot of the given one from the perpendicular, but on the other side of it. Let A be the given point, BC the given straight line ; let AD be the perpendicular from A on BC, and AE, AF any two obliques of which AF makes the greater angle with AD. Produce AD to A', making A'D equal to AD, and join A'E, A F. Then the triangles ADE, A'DE are equal in all respects ; and so are the triangles ADF, A'DF. Now (1) in the triangle AEA' the two sides AE, EA' are-greater than AA' [1. 20I, that is, twice AE is greater than twice AD. a 9 * BOOK I [l. a i, Therefore AE is greater than AD. (a) Since AE, A'E are drawn to E, a point within the triangle A FA', AE, EA' are together greater than AE, EA\ [l *i] or twice AE is greater than twice AE.. Therefore AE is greater than AE. (3) Along DB measure off DG equal to DF, and join AG. The triangles AGD, AFD are then equal in all respects, so that the angles GAD, FAD are equal, and AG is equal to AF. Proposition 22. Out of three straight lines, which are equal to three given straight lines, to construct a triangle : thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one. [1. *o] Let the three given straight lines be A, B, C, and of these let two taken together in any manner be greater than the remaining one, namely A, B greater than C, A, C greater than B, and B, C greater than A ; thus it is required to construct a triangle out of straight lines equal to A, B, C. a- 8- c- Let there be set out a straight line DB, terminated at D but of infinite length in the direction of B, and let DF be made equal to A, FG equal to B, and GH equal to C. [1. 3] With centre F and distance FD let the circle DKL be described ; again, with centre G and distance GH let the circle KLH be described ; and let KF, KG be joined ; I say that the triangle KFG has been constructed out of three straight lines, equal to A, B, C. i. 11] PROPOSITIONS a i, a* »93 For, since the point F is the centre of the circle DKS,, FD is equal to FK. But FD is equal to A ; therefore KF is also equal to A. Again, since the point G is the centre of the circle LKH, GH is equal to GK. But GH is equal to C; therefore KG is also equal to C. And FG is also equal to B ; therefore the three straight lines KF, FG, GK are equal to the three straight lines A, B, C. Therefore out of the three straight lines KF, FG, GK, which are equal to the three given straight lines A, B, C, the triangle KFG has been constructed. 6 q. e. F. i — *> This is the first cast in the Elements of a Satparfiii to a problem in the sense of a statement of the conditions or limits of the possibility of a solution. The criterion is of course supplied by the preceding proposition. j. thu» It is necessary. This is usually translated (e.g. by Williamson and Simson) "But it is necessary," which is however inaccurate, since the Greek is not Sti W but &ri ti). The words are the same as those used to introduce the ttopurpit in the other sense of the " definition " or " particular statement " of a construction to be effected. Hence, as in the latter case we say " thus it is required " (e.g. to bisect the finite straight line AS, I. 10}, we should here translate " thus it is necessary. 4* To this enunciation alt the M5S. and Bocthius add, after the Sioptefifa, the words "because in any triangle two sides taken together in any manner are greater than the remaining one." But this explanation has the appearance of a gloss, and it is omitted hy Proclus and Campanus. Moreover there is no corresponding addition to the Supwitit of vi, 18. It was early observed that Euclid assumes, without giving any reason, that the circles drawn as described will meet if the condition that any two of the straight lines A, S, C are together greater than the third be fulfilled. Prod us (p. 33 r, S sqq.) argues the matter by means of redudio ad abrurdum, but does not exhaust the possible hypotheses inconsistent with the contention. He says the circles must do one of three things, (1) cut one another, (1) touch one another, (3} stand apart {&«rrarai) from one another. He then considers the hypotheses (a) of their touching externally, (t>) of their being separated from one another by a space. He should have considered also the hypothesis (r) of one circle touching the other internally or lying entirely within the other without touching. These three hypotheses being successively disproved, it follows that the circles must meet (this is the line taken by Camerer and Todhunter). Simson says : " Some authors blame Euclid because he does not demonstrate that the two circles made use of in the construction of this problem must cut one another : but this is very plain from the determination he has given, namely, that any two of the straight lines DF, FG, GH must be greater than the third, For who is so dull, though only beginning to learn the Elements, as not to perceive that the circle described from the centre F, at the distance FD, must meet FH betwixt F and H, because FD is less than FH; and that, for the like reason, the circle described from the ig4 BOOK I [i. 22, 33 centre G at the distance GH must meet DG betwixt D and G ; and that these circles must meet one another, because FD and GH are together greater than FG." We have in fact only to satisfy ourselves that one of the circles, e.g. that with centre G, has at least one point of its circumference outside the other circle and also at least one point of its circumference inside the same circle ; and this is best shown with reference to the points in which the first circle cuts the straight line DE. For (i) FH, being equal to the sum of B and C, is greater than A t i.e. than the radius of the circle with centre F, and therefore His outside that circle. (2) If GAf be measured along GF equal to GH or C, then, since GM is either (a) less or {p} greater than GF, Jfwill fall either {a) between G and F or (6) beyond F towards D ; in the first case (a) the sum of FM and C is equal to FG and therefore less than the sum of A and C, so that FM is less than A or FD ; in the second case {/>) the sum of MF and FG, i.e. the sum of MF and B, is equal to GAf or C, and therefore less than the sum of ^ and B, so that MF is less than A or FD ; hence in either case M falls within the circle with centre F. It being now proved that the circumference of the circle with centre G has at least one point outside, and at least one point inside, the circle with centre F, we have only to invoke the Principle of Continuity, as we have to do in 1. 1 (cf. the note on that proposition, p. 242, where the necessary postulate is stated in the form suggested by Killing). That the construction of the proposition gives only two points of intersection between the circles, and therefore only two triangles satisfying the condition, one on each side of FG, is clear from I. 7, which, as before pointed out, takes the place, in Book 1., of 111. 10 proving that two circles cannot intersect in more points than two. Proposition 23. On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle. Let AB be the "given straight line, A the point on it, and the angle DCE the given rectilineal angle; thus it is required to construct on the given straight line AB, and at the point A on it, a rectilineal angle equal to the given rectilineal angle DCE. On the straight lines CD, CE respectively let the points D, E be taken at random ; let DE be joined, and out of three straight lines which are equal to the three li »3] PROPOSITIONS 22, 23 395 straight lines CD, DE, CE let the triangle AFG he. con- structed in such a way that CD is equal to AF, CE to AG, and further DE to FG. [1. 22] Then, since the two sides DC, CE are equal to the two sides FA., AG respectively, and the base DE is equal to the base FG, the angle DCE is equal to the angle FAG. [1. 8] Therefore on the given straight line A3, and at the point A on it, the rectilinealangle FAG has been constructed equal to the given rectilineal angle DCE. e Q. E. F. This problem was, according to Eudemus (see Proclus, p. 333, 5), "rather the discovery of Oenopides," from which we must apparently infer, not that Oenopides was the first to find any solution of it, but that it was he who dis- covered the particular solution given by Euclid. (Cf. Bretschneider, p. 65.) The editors do not seem to have noticed the fact that the construction of the triangle assumed in this proposition is not exactly the construction given in 1. 22. We have here to construct a triangle on a certain finite straight line AG as base; in 1. 11 we have only to construct a triangle with sides of given length without any restriction as to how it is to be placed. Thus in 1. ■• we set out any tine whatever and measure successively three lengths along it beginning from the given extremity, and what we must regard as the base is the intermediate length, not the length beginning at the given extremity, of the straight line arbitrarily set out. Here the base is a given straight line abutting at a given point Thus the construction has to be modified somewhat from h B that of the preceding proposition. We must measure AG along AB so that AG is equal to CE (or CD), and GH along GB equal to DE; and then we must produce BA, in the opposite direction, to F, so that AF'is equal to CD (or CE, if AG has been made equal to CD). Then, by drawing circles (1) with centre A and radius AF, (2) with centre G and radius GH, we determine K, one of their points of intersection, and we prove that the triangle KAG is equal in all respects to the triangle DCE, and then that the angle at A is equal to the angle DCE. I think that Proclus must (though he does not say so) have felt the same difficulty with regard to the use in 1. 33 of the result of 1. 22, and that this is probably the reason why he gives over again the construction which I have given above, with the remark (p. 334, 6) that "you may obtain the construction of the triangle in a more instructive manner (StSatricaA.iKioTi^oi') as follows," Proclus objects to the procedure of Apollonius in constructing an angle under the same conditions, and certainly, if he quotes Apollonius correctly, the tatter's exposition must have been somewhat slipshod. 296 BOOK I {]. *3, «4 "He takes an angle CDE at random," says Prod us (p. 335> '9 s Vl-)> ' ,and a straight line AB, and with centre D and distance CD describes the circumference CE, and in the same way with centre A and distance AB the circumference FB. Then, cutting off FB equal to CE, he joins AF. And he declares that the angles A, D standing on equal circumferences are equal." In the first place, as Prod us remarks, it should be premised that AB is equal to CD in order that the circles may be equal; and the use of Book lit. for such an elementary construction is objectionable. The omission to state that AB must be taken equal to CD was no doubt a slip, if it occurred. And, as regards the equal angles "standing on equal circum- ferences," it would seem possible that Apollonius said this in explanation, for the sake of brevity, rather than by way of proof. It seems to me probable that his construction was only given from the point of view of practical, not theoretical, geometry. It really comes to the same thing as Euclid's except that DC is taken equal to DE. For cutting off the arc BF equal to the arc CE can only be meant in the sense of measuring the chord CE, say, with a pair of compasses, and then drawing a circle with centre B and radius equal to the chord CE. Apollonius' direction was therefore probably intended as a practical short cut, avoiding the actual drawing of the chords CE, BF, which, as well as a proof of the equality in all respects of the triangles CDE, BAF, would be required to establish theoretically the correct- ness of the construction. Proposition 24. If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the equal straight lines greater than the other, they will also have the base greater than the base. 5 Let ABC, DEF be two triangles having the two sides AB, A C equal to the two sides DE, DF respectively, namely AB to DE, and A C to DF, and let the angle at A be greater than the angle at D ; I say that the base BC is also greater than the base £F, 10 For, since the angle BAC is greater than the angle EDF, let there be constructed, on the straight line DE, and at the point D on it, the angle EDG jj equal to the angle BAC; [1. 13] let DG be made equal to either of the two straight lines AC, DF, and let EG, FG be joined. i. 24] PROPOSITIONS 23, 24 *97 Then, since AB is equal to DE, and AC to DG, *>the two sides BA, AC are equal to the two sides ED, DG, respectively ; and the angle BAC is equal to the angle EDG ; therefore the base BC is equal to the base EG. [1. 4] Again, since DF is equal to DG, 25 the angle DGF is also equal to the angle DFG ; [l 5] therefore the angle DFG is greater than the angle EGF. Therefore the angle EFG is much greater than the angle EGF. And, since EFG is a triangle having the angle EFG 30 greater than the angle EGF, and the greater angle is subtended by the greater side, [*. 19] the side EG is also greater than EF. But EG is equal to BC. Therefore BC is also greater than EF. 3S Therefore etc. Q. E. D. 10. I have naturally left out the well-known words added by Simson in order to avoid the necessity of considering three cases : " Of the two sides DE, DF let DE be the side which is not greater than the other." I doubt whether Euclid could have been induced to insert the words himself, even if it had been represented to him that their omission meant leaving two possible cases out of consideration. His habit and that of the great Greek geometers was, not to set out all possible cases, but to give as a rule one case, generally the most difficult, as here, and to leave the others to the reader to work out for himself. We have already seen one instance in 1. 7. Proclus of course gives the other two cases which arise if we do not first provide that DE is not greater than DF. (1) In the first case G may fall on EF produced, and it is then obvious that EG is greater than EF. (2) In the second case EG may fall below EF. If so, by 1. 21, DF, FE are together less than DG, GE. But DF is equal to DG ; there- fore EF is less than EG, i.e. than BC. These two cases are therefore decidedly simpler than the case taken by Euclid as typical, and could well be left to the ingenuity of the learner. If however after all we prefer to insert Simson's words and avoid the latter BOOK I [i-*4 two cases, the proof is not complete unless we show that, with his assumption, /"must, in the figure of the proposition, fall below EG. De Morgan would make the following proposition precede: Every straight tine drawn from the vertex of a triangle to the base is less than the greater of the two sides, or than either if they are equal, and he would prove it by means of the proposition relating to perpendicular and obliques given above, p. 291. But it is easy to prove directly that F falls below EG, if DE is not greater than DG, by the method employed by Pfleiderer, Lardner, and Todhunter. Let DF, produced if necessary, meet EG in H. Then the angle DUG is greater than the angle DEG\ ["• >6] and the angle DEG is not less than the angle DGE ; [.. .8] therefore the angle DUG is greater than the angle DGH. Hence DH is less than DG, [1. 19] and therefore DH is less than DF. Alternative proof. Lastly, the modern alternative proof is worth giving. A Let DHhisect the angle FDG (after the triangle DEG has been made equal in all respects to the triangle A BC t as in the proposition), and let DH meet EG in H. Join HF. Then, in the triangles FDH, GDH, the two sides FD, DH are equal to the two sides GD, DH, and the included angles FDH, GDH at equal ; therefore the base HF is equal to the base HG Accordingly EG is equal to the sum of EH, HF; and EH, HF are together greater than EF; [1. 20] therefore EG, or BC, is greater than EF. Proclus (p. 339, 1 1 sqq.) answers by anticipation the possible question that might occur to any one on this proposition, viz. why does Euclid not compare the areas of the triangles as he does in 1. 4 ? He observes that inequality of the areas does not follow from the inequality of the angles contained by the equal sides, and that Euclid leaves out all reference to the question both for this reason and because the areas cannot be compared without the help of the theory of parallels. " But if," says Proclus, " we must anticipate what is to come and make our comparison of the areas at once, we assert that (1) if the angles A, D — supposing that our argument proceeds with reference to the figure in the proposition — are {together) equal to two right angles, the triangles L 34, 25] PROPOSITIONS 14, 25 299 a« primed equal, {2) if greater than tins right angles, that triangle which has the greater angle is less, and (3) </ they are less, greater," Proclus then gives the proof, but without any reference to the source from which he quoted the proposition. Now an-NairizI adds a similar proposition to 1. 38, but definitely attributes it to Heron. I shall accordingly give it in the place where Heron put it Proposition 25. If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other. Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE, and A C to DF ; and let the base BC be greater than the base FF; I say that the angle BAC is also greater than the angle EDF For, if not, it is either equal to it or less. Now the angle BAC is not equal to the angle EDF; for then the base BC would also have been equal to the base EF r> 4] but it is not ; therefore the angle BA C is not equal to the angle EDF. Neither again is the angle BA C less than the angle EDF; for then the base BC would also have been less than the base EF, [ft 24] but it is not ; therefore the angle BA C is not less than the angle EDF. But it was proved that it is not equal either ; therefore the angle BAC is greater than the angle EDF. Therefore etc. Q. E. D. 3 oo BOOK I [i. 15 De Morgan points out that this proposition (as also i. 8) is a purely logical consequence of i. 4 and 1. 14 in the same way as 1. r 9 and 1. 6 are purely logical consequences of 1. 18 and 1. 5. If d, 6, e denote the sides, A, B, C the angles opposite to them in a triangle ABC, and a', b', /, A', E, C the sides and opposite angles respectively in a triangle A'HC, 1. 4 and I. 14 tell us that, i, e being respectively equal to i\ /, ( 1 ) if A is equal to A\ then a is equal to a', (z) if A is less than A', then a is less than «', (3) if A is greater than A', then a is greater than a' ; and it follows logically that, (1) if a is equal to a, the angle A is equal to the angle A', [1. 8] (3) if a is less than a, A is less than A', \ (3) if a is greater than d, A is greater than A'. } I * a 5J Two alternative proofs of this theorem are given by Proclus (pp. 345 — 7), and they are both interesting. Moreover both are direct. I. Proof by Menelaus of Alexandria. Let ABC, DEB" be two triangles having the two sides BA, AC equal to the two sides ED, DF, but the base BC greater than the base EF. Then shall the angle at A be greater than the angle at D. From BC cut off BG equal to EF. At B, on the straight line BC, make the angle GBH (on the side of BG remote from A) equal to the angle FED. Make BH equal to DE ; join HG, and produce it to meet A C in K. Join AH. Then, since the two sides GB, BH are equal to the two sides FE, ED respectively, and the angles contained by them are equal, HG is equal to ZVor AC, and the angle BHG is equal to the angle EDF. Now HK is greater than HG or AC, and a fortiori greater than AK; therefore the angle KAH\% greater than the angle KHA. And, since AB is equal to BH, the angle BAH is equal to the angle BHA. Therefore, by addition, the whole angle BA C is greater than the whole angle BHG, that is, greater than the angle EDF. i. «5, a6] PROPOSITIONS 35, 36 301 II, Heron's proof. Let the triangles be given as before. Since BC is greater than EF, produce EF to G so that EG is equal to EC. Produce ED to If so that DH is equal to DF. The circle with centre D and radius DFwiW then pass through H. Let it be described, as FKH. Now, since BA, AC are together greater than £C, and Atf, ^C are equal to ,££>, Z>Zf respectively, while JC is equal to EG, EH is greater than EG. Therefore the circle with centre E and radius EG will cut Elf, and therefore will cut the circle already drawn. Let it cut that circle in K, and join DK, KB. Then, since D is the centre of the circle FKH, DK is equal to Z>^or AC Similarly, since £ is the centre of the circle KG, EK is equal to EG or BC, And DE is equal to A B. Therefore the two sides BA, AC are equal to the two sides ED, DK respectively; and the base BC is equal to the base EK; therefore the angle BAC is equal to the angle EDK Therefore the angle BA C is greater than the angle EDF. Proposition 26, If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending erne 0/ the equal angles, they wilt also have the remaining sides equal to s the remaining sides and the remaining angle to the remaining angle. 3°* BOOK I [1. 26 Let ABC, DEF be two triangles having the two angles ABC, BCA equal to the two angles DEF, EFD respectively, namely the angle ABC to the angle DEF, and the angle 10 BCA to the angle EFD ; and let them also have one side equal to one side, first that adjoining the equal angles, namely BC to EF; I say that they will also have the remaining sides equal to the remaining sides respectively, namely AB to DE and >s AC to DF, and the remaining angle to the remaining angle, namely the angle BA C to the angle EDF. For, if AB is unequal to DE, one of them Is greater. Let AB be greater, and let BG be made equal to DE ; and let GC be joined. 20 Then, since BG is equal to DE, and BC to EF, the two sides GB, BC are equal to the two sides DE, EF respectively; and the angle GBC is equal to the angle DEF ; therefore the base GC is equal to the base DF, 25 and the triangle GBC is equal to the triangle DEF, and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend ; [1. 4] therefore the angle GCB is equal to the angle DFE. But the angle DFE is by hypothesis equal to the angle BCA; 3 o therefore the angle BCG is equal to the angle BCA, the less to the greater : which is impossible. Therefore AB is not unequal to DE, and is therefore equal to it. But BC is also equal to EF; 35 therefore the two sides AB, BC are equal to the two sides DE, EF respectively, and the angle ABC is equal to the angle DEF; therefore the base A C is equal to the base DF, and the remaining angle BAC is equal to the remaining 40 angle EDF. [l 4] I. *6] PROPOSITION a* 303 Again, let sides subtending equal angles be equal, as AB to DE; I say again that the remaining sides will be equal to the remaining sides, namely AC to DF and BC to EF, and 45 further the remaining angle BAC is equal to the remaining angle EDF. For, if BC is unequal to EF, one of them is greater. Let BC be greater, if possible, and let BH be made equal to EF; let AH be joined. 50 Then, since BH is equal to EF, and AB to DE, the two sides AB, BH are equal to the two sides DE, EF respectively, and they contain equal angles ; therefore the base AH is equal to the base DF, and the triangle ABH is equal to the triangle DEF, ss and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend ; [1. 4] therefore the angle BHA is equal to the angle EFD. But the angle EFD is equal to the angle BCA ; therefore, in the triangle AHC, the exterior angle BHA is 60 equal to the interior and opposite angle BCA : which is impossible. [1. 16] Therefore BC is not unequal to EF, and is therefore equal to it. But AB is also equal to DE ; 65 therefore the two sides AB, BC are equal to the two sides DE, EF respectively, and they contain equal angles ; therefore the base AC is equal to the base DF, the triangle ABC equal to the triangle DEF, and the remaining angle BAC equal to the remaining angle r>EDF, [1.4] Therefore etc. Q, E. D. 1 — 3. the aide adjoining the equal angles, rhtupir Hjr rpot roTi but ywrtati. 39. la by hypothesis equal. vr6Ktir at tnj, according to the elegant Greek idiom. ifbtettiat is used for the passive of irrvrlffitfn, as Keit*eu is used for the passive of riBrtfu, and to with the other compounds. Cf. Trpxruiitidat, " to be added." The alternative method of proving this proposition, viz. by applying one triangle to the other, was very early discovered, at least so far as regards the case where the equal sides are adjacent to the equal angles in each. An-Nairizi gives it for this case, observing that the proof is one which he had found, but of which he did not know the author. 3o 4 BOOK I [i 3 6 Proclus has the following interesting note {p. 35*, 13 — 18): "Eudemus in his geometrical history refers this theorem to Thales. For he says that, in the method by which they say that Thales proved the distance of ships in the sea, it was necessary to make use this theorem." As, unfortunately, this information is not sufficient of itself to enable us to determine how Thales solved this problem, there is considerable room for conjecture as to bis method. The suggestions of Bretschneider and Cantor agree in the assumption that the necessary observations were probably made from the top of some tower or structure of known height, and that a right-angled triangle was used in which the tower was the perpendicular, and the line connecting the bottom of the tower and the ship was the base, as in the annexed figure, where AB is the tower and C the ship. Bretschneider (Die Geometrie and die Geometer vor Eukleides, § 30) says that it was only necessary for the observer to observe the angle CAB, and then the triangle would be completely determined by means of this angle and the known length AB. As Bretschneider says that the result would be obtained "in a moment " by this method, it is not clear in what sense he supposes Thales to have "observed" the angle SAC. Cantor is more definite (GescA. d. Math. i ( , p. 145), for he says that the problem was nearly related to that of finding the Seqt from given sides. By the Seqt in the Papyrus Rhind is meant the ratio to one another of certain lines in pyramids ot obelisks. Eisenlohr and Cantor took the one word to be equivalent, sometimes to the cosine of the angle made by the edge of the pyramid with the coterminous diagonal of the base, sometimes to the tangent of the angle of slope of the faces of the pyramid. Ft is now certain that it meant one thing, viz. the ratio of half the side of the base to the height of the pyramid, i.e. the cotangent of the angle of slope. The calculation of the Seqt thus implying a sort of theory of simi- larity, or even of trigonometry, the suggestion of Cantor is apparently that the Seqt in this case would be found from a smalt right-angled triangle ADE having a common angle A with ABC as shown in the figure, and that the ascertained value of the Stqt with the length AB would determine BC. This amounts to the use of the property of similar triangles ; and Bretschneider's suggestion must apparently come to the same thing, since, even if Thales measured the angle in our sense (e.g. by its ratio to a right angle), he would, in the absence of something corresponding to a table of trigonometrical ratios, have gained nothing and would have had to work out the proportions all the same. Max C P, Schmidt also (Kulturhistorische Beiirage zur Kenntnis des griechisehcn und romischen Alter turns, 1906, p. 32) similarly supposes Thales to have had a right angle made of wood or bronze with the legs graduated, to have placed it in the position ADE (A being the position of his eye}, and then to have read off the lengths AD, DE respectively, and worked out the length of BC by the rule of three. How then does the supposed use of similar triangles and their property square with Eudemus' remark about 1. 26 ? As it stands, it asserts the equality of two triangles which have two angles and one side respectively equal, and the theorem can only be brought into relation with the above explanations by taking it as asserting that, if two angles and one side of one triangle are given, the triangle is completely determined. But, if Thales i. 26] PROPOSITION 26 305 practically used proportions, as supposed, 1. 26 is surely not at all the theorem which this procedure would naturally suggest as underlying it and being "necessarily used"; the use of proportions or of similar but not equal triangles would surely have taken attention altogether away from 1. 26 and fixed it on vi. 4. For this reason I think Tannery is on the right road when he tries to find a solution using 1. 26 as it stands, and withal as primitive as any recorded solution of such a problem. His suggestion (La Gtemitrit gretqut, pp. 90—1) is based on the fiuminis varatio of the Roman agrimensor Marcus Junius Nipsus and is as follows. To find the distance from a point A to an inaccessible point B. From A measure along a straight line at right angles to AB a length AC and bisect it at D. From C draw C£ at right angles to CA on the side of it remote from B, and let £ be the point on it which is in a straight line with B and D. Then, by 1. 26, CE is obviously equal to AB. As regards the equality of angles, it is to be observed that those at D are equal because they are vertically opposite, and, curiously enough, Thales is expressly credited with the discovery of the equality of such angles. The only objecti